Please  Note:  This  item  is  subject  to 
it  CALL  after  one  week. 

_PATI 


"# 

CALIFO^^V 


SAE 


3   1822  01155  6347 

CIFT  Di    <v 

DR.  GEORGE  F.  McEWEN 


Columbia  Onitoergitp  3lecture£ 


GRAPHICAL  METHODS 

ERNEST  KEMPTON  ADAMS  RESEARCH  FUND 
1909-1910 


COLUMBIA 

UNIVERSITY  PRESS 
SALES  AGENTS 

NEW  YORK  : 

LEMCKE  &  BUECHNER 
30-32  WEST  2?TH  STREET 

LONDON  : 

HENRY  FROWDE 
AMES  CORNER,  E.G. 

TORONTO : 

HEXRY  FROWDE 

25  RICHMOND  STREET,  W. 


COLUMBIA    UNIVERSITY  LECTURES 


GRAPHICAL  METHODS 


BY 


CARL  RUNGE,  PH.D. 

PROFESSOR    OF   APPLIED   MATHEMATICS   IN    THE    UNIVERSITY    OP   GOTTINGEN 

KAISER  WILHELM  PROFESSOR  OF  GERMAN  HISTORY  AND  INSTITUTIONS 

FOB  THE  TEAR   1909-1910 


JQeto 

COLUMBIA  UNIVERSITY  PRESS 
1912 


LANCASTER.  PA. 


INTRODUCTION. 

§  1.  A  great  many  if  not  all  of  the  problems  in  mathematics 
may  be  so  formulated  that  they  consist  in  finding  from  given 
data  the  values  of  certain  unknown  quantities  subject  to  certain 
conditions.  We  may  distinguish  different  stages  in  the  solution 
of  a  problem.  The  first  stage  we  might  say  is  the  proof  that  the 
quantities  sought  for  really  exist,  that  it  is  possible  to  satisfy 
the  given  conditions  or,  as  the  case  may  be,  the  proof  that  it  is 
impossible.  In  the  latter  case  we  have  done  with  the  problem. 
Take  for  instance  the  celebrated  question  of  the  squaring  of  the 
circle.  We  may  in  a  more  generalized  form  state  it  thus:  Find 
the  integral  numbers,  which  are  the  coefficients  of  an  algebraic 
equation,  of  which  TT  is  one  of  the  roots.  Thirty  years  ago 
Lindemann  showed  that  integral  numbers  subject  to  these  con- 
ditions do  not  exist  and  thus  a  problem  as  old  almost  as 
human  history  came  to  an  end.  Or  to  give  another  instance 
take  Fermat's  problem,  for  the  solution  of  which  the  late  Mr. 
Wolfskehl,  of  Darmstadt,  has  left  $25,000  in  his  will.  Find  the 
integral  numbers  x,  y,  z  that  satisfy  the  equation 


where  n  is  an  integral  number  greater  than  two.  Fermat  main- 
tained that  it  is  impossible  to  satisfy  these  conditions  and  he  is 
probably  right.  But  as  yet  it  has  not  been  shown.  So  the 
solution  of  the  problem  may  or  may  not  end  in  its  first  stage. 
In  many  other  cases  the  first  stage  of  the  solution  may  be  so 
easy,  that  we  immediately  pass  on  to  the  second  stage  of  finding 
methods  to  calculate  the  unknown  quantities  sought  for.  Or 
even  if  the  first  stage  of  the  solution  is  not  so  easy,  it  may  be 
expedient  to  pass  on  to  the  second  stage.  For  if  we  succeed  in 
finding  methods  of  calculation  that  determine  the  unknown  quan- 


VI  GRAPHICAL   METHODS. 

titles,  the  proof  of  their  existence  is  included.  If  on  the  other 
hand,  we  do  not  succeed,  then  it  will  be  time  enough  to  return 
to  the  first  stage. 

There  are  not  a  small  number  of  men  who  believe  the  task  of 
the  mathematician  to  end  here.  This,  I  think,  is  due  to  the 
fact  that  the  pure  mathematician  as  a  rule  is  not  in  the  habit  of 
pushing  his  investigation  so  far  as  to  find  something  out  about  the 
real  things  of  this  world.  He  leaves  that  to  the  astronomer,  to 
the  physicist,  to  the  engineer.  These  men,  on  the  other  hand, 
take  the  greatest  interest  in  the  actual  numerical  values  that 
are  the  outcome  of  the  mathematical  methods  of  calculation. 
They  have  to  carry  out  the  calculation  and  as  soon  as  they  do  so, 
the  question  arises  whether  they  could  not  get  at  the  same  result 
in  a  shorter  way,  with  less  trouble.  Suppose  the  mathematician 
gives  them  a  method  of  calculation  perfectly  logical  and  con- 
clusive but  taking  200  years  of  incessant  numerical  work  to 
complete.  They  would  be  justified  in  thinking  that  this  is  not 
much  better  than  no  method  at  all.  So  there  arises  a  third  stage 
of  the  solution  of  a  mathematical  problem  in  which  the  object  is 
to  develop  methods  for  finding  the  result  with  as  little  trouble  as 
possible.  I  maintain  that  this  third  stage  is  just  as  much  a 
chapter  of  mathematics  as  the  first  two  stages  and  it  will  not  do 
to  leave  it  to  the  astronomer,  to  the  physicist,  to  the  engineer  or 
whoever  applies  mathematical  methods,  for  this  reason  that 
these  men  are  bent  on  the  results  and  therefore  they  will  be  apt 
to  overlook  the  full  generality  of  the  methods  they  happen  to 
hit  on,  while  in  the  hands  of  the  mathematician  the  methods 
would  be  developed  from  a  higher  standpoint  and  their  bearing 
on  other  problems  in  other  scientific  inquiries  would  be  more 
likely  to  receive  the  proper  attention. 

The  state  of  affairs  today  is  such  that  in  a  number  of  cases  the 
methods  of  the  engineer  or  the  surveyor  are  not  known  to  the 
astronomer  or  the  physicist,  or  vice  versa,  although  their  prob- 
lems may' be  mathematically  almost  identical.  It  is  particularly 
so  with  graphical  methods,  that  have  been  invented  for  definite 


INTRODUCTION.  Vll 

problems.  A  more  general  exposition  makes  them  applicable 
to  a  vast  number  of  cases  that  were  originally  not  thought  of. 
In  this  course  I  shall  review  the  graphical  methods  from  a 
general  standpoint,  that  is,  I  shall  try  to  formulate  and  to  teach 
them  in  their  most  generalized  form  so  as  to  facilitate  their 
application  in  any  problem,  with  which  they  are  mathematically 
connected.1  The  student  is  advised  to  do  practical  exercises. 
Nothing  but  the  repeated  application  of  the  methods  will  give 
him  the  whole  grasp  of  the  subject.  For  it  is  not  sufficient  to 
understand  the  underlying  ideas,  it  is  also  necessary  to  acquire  a 
certain  facility  in  applying  them.  You  might  as  well  try  to  learn 
piano  playing  only  by  attending  concerts  as  to  learn  the 
graphical  methods  only  through  lectures. 

1  For  the  literature  of  the  subject  see  "  Encyklopadie  der  mathematischen 
Wissenschaften,"  Art.  R.  Mehmke,  "  Numerisches  Rechnen,"  and  Art.  F 
Willers  and  C.  Runge,  "Graphische  Integration." 


TABLE  OF  CONTENTS. 
1.  Introduction.  . 


CHAPTER  I.     Graphical  Calculation. 

2.  Graphical  arithmetic 1 

3.  Integral  functions 6 

4.  Linear  functions  of  any  number  of  variables 18 

5.  The  graphical  handling  of  complex  numbers 25 


CHAPTER  II.     The  Graphical  Representation  of  Functions  of  One 
or  More  Independent  Variables 

§    6.  Functions  of  one  independent  variable 40 

§    7.  The  principle  of  the  slide  rule 43 

§    8.  Rectangular  coordinates  with  intervals  of  varying  size  52 

§    9.  Functions  of   two    independent   variables 58 

§  10.  Depiction  of  one  plane  on  another  plane    .  .  65 

§  11.  Other  methods  of  representing  relations  between  three 

variables 84 

§  12.  Relations  between  four  variables 94 

CHAPTER  III.     The  Graphical  Methods  of  the  Differential  and 
Integral  Calculus. 

§  13.  Graphical  integration 101 

§  14.  Graphical  differentiation 117 

§  15.  Differential  equations  of  the  first  order 120 

§  16.  Differential  equations  of  the  second  and  higher  orders  136 


CHAPTER  I. 
GKAPHICAL  CALCULATION. 

§  2.  Graphical  Arithmetic. — Any  quantity  susceptible  of  mensu- 
ration can  be  graphically  represented  by  a  straight  line,  the 
length  of  the  line  corresponding  to  the  value  of  the  quantity. 
But  this  is  by  no  means  the  only  possible  way.  A  quantity 
might  also  be  and  is  sometimes  graphically  represented  by  an 
angle  or  by  the  length  of  a  curved  line  or  by  the  area  of  a  square 
or  triangle  or  any  other  figure  or  by  the  anharmonic  ratio  of  four 
points  in  a  straight  line  or  in  a  variety  of  other  ways.  The 
representation  by  straight  lines  has  some  advantages  over  the 
others,  mainly  on  account  of  the  facility  with  which  the  ele- 
mentary mathematical  operations  can  be  carried  out. 

What  is  the  use  of  representing  quantities  on  paper?  It  is  a 
convenient  way  of  placing  them  before  our  eye,  of  comparing 
them,  of  handling  them.  If  pencil  and  paper  were  not  as  cheap 
as  they  are,  or  if  to  draw  a  line  were  a  long  and  tedious  under- 
taking, or  if  our  eye  were  not  as  skillful  and  expert  an  assistant, 
graphical  methods  would  lose  much  of  their  significance.  Or, 
on  the  other  hand,  if  electric  currents  or  any  other  measurable 
quantities  were  as  cheaply  and  conveniently  produced  in  any 
desired  degree  and  added,  subtracted,  multiplied  and  divided 
with  equal  facility,  it  might  be  profitable  to  use  them  for  the 
representation  of  any  other  measurable  quantities,  not  so  easily 
produced  or  handled. 

The  addition  of  two  positive  quantities  represented  by  straight 
lines  of  given  length  is  effected  by  laying  them  off  in  the  same 
direction,  one  behind  the  other.  The  direction  gives  each  line  a 
beginning  and  an  end.  The  beginning  of  the  second  line  has  to 
coincide  with  the  end  of  the  first,  and  the  resulting  line  represent- 
ing the  sum  of  the  two  runs  from  the  beginning  of  the  first  to 
2  1 


GRAPHICAL  METHODS. 


the  end  of  the  second.  Similarly  the  subtraction  of  one  positive 
quantity  from  another  is  effected  by  giving  the  lines  opposite  direc- 
tions and  letting  the  beginning  of  the  line  that  is  to  be  subtracted 
coincide  with  the  end  of  the  other.  The  result  of  the  subtrac- 
tion is  represented  by  the  line  that  runs  from  the  beginning  of 
the  minuend  to  the  end  of  the  subtrahend.  The  result  is  positive 
when  this  direction  coincides  with  that  of  the  minuend,  and  nega- 
tive when  it  coincides  with  that  of  the  subtrahend.  This  leads 
to  the  representation  of  positive  and  negative  quantities  by  lines  of 
opposite  direction.  The  subtraction  of  one  positive  quantity  from 
another  may  then  be  looked  upon  as  the  addition  of  a  positive  and 
a  negative  quantity.  I  do  not  want  to  dwell  on  the  logical  explana- 
tion of  this  subject,  but  I  want  to  point  out  the  practical  method 
used  for  adding  a  large  number  of  positive  and  negative  quantities 
represented  by  straight  lines  of  opposite  direction.  Take  a 
straight  edge,  say  a  piece  of  paper  folded  over  so  as  to  form  a 
straight  edge,  mark  a  point  on  it,  and  assign  one  of  the  two 
directions  as  the  positive  one.  Lay  the  edge  in  succession  over 
the  different  lines  and  run  a  pointer  along  it  through  an  amount 
equal  in  each  case  to  the  length  of  the  line  and  in  the  positive 
or  negative  direction  according  to  the  sign  of  the  quantity.  The 
pointer  is  to  begin  at  the  point  marked.  The  line  running  from 
this  point  to  where  the  pointer  stops  represents  the  sum  of  the 

given  quantities.  The  advan- 
tage of  this  method  is  that  the 
intermediate  positions  of  the 
pointer  need  not  be  marked  pro- 
vided only  that  the  pointer  keeps 
its  position  during  the  move- 
'  ment  of  the  edge  from  one  line 
to  the  next.  As  an  example  take 
the  area,  Fig.  1.  A  number  of 

rectangular  strips  j  cm.  wide  are  substituted  for  the  area  so  that, 
measured  in  square  centimeters,  it  is  equal  to  half  the  sum  of 
the  lengths  of  the  strips  measured  in  centimeters.  The  straight 


FIG  1. 


GRAPHICAL  CALCULATION.  3 

edge  is  placed  over  the  strips  in  succession  and  the  pointer  is 
run  along  them.  The  edge  is  supposed  to  carry  a  centime- 
ter scale  and  the  pointer  is  to  begin  at  zero.  The  final  position 
of  the  pointer  gives  half  the  value  of  the  area  in  square  centi- 
meters. The  drawing  of  the  strips  may  be  dispensed  with,  their 
lengths  being  estimated,  only  their  width  must  be  shown.  If 
the  scale  should  be  too  short  for  the  whole  length,  the  only  thing 
we  have  to  do  is  to  break  any  of  the  lengths  that  range  over  the 
end  of  the  scale  and  to  count  how  many  times  we  have  gone 
over  the  whole  scale.  I  have  found  it  convenient  to  use  a  little 
pointer  of  paper  fastened  on  the  runner  of  a  slide  rule  so  that  it 
can  be  moved  up  and  down  the  metrical  scale  on  one  side  of  the 


FIG.  2. 


slide  rule.  The  area  is  in  this  manner  determined  rapidly  and 
with  considerable  accuracy,  very  well  comparable  to  the  ac- 
curacy of  a  good  planimeter.  If  the  area  of  any  closed  curve 
is  to  be  found,  the  way  to  proceed  is  to  choose  two  parallel 
lines  that  cut  off  two  segments  on  either  side  (see  Fig.  2),  to 
measure  the  area  between  them  by  the  method  described  above 
and  to  estimate  the  two  segments  separately.  If  the  curves  of 
the  segments  may  with  sufficient  accuracy  be  regarded  as  arcs 
of  parabolas  the  area  would  be  two  thirds  the  product  of  length 
and  width.  If  not  they  would  have  to  be  estimated  by  substitut- 
ing a  rectangle  or  a  number  of  rectangles  for  them. 


GEAPHICAL  METHODS. 


In  the  same  way  the  addition  and  subtraction  of  pure  numbers 
may  also  be  carried  out.  We  need  only  represent  the  numbers 
by  the  ratios  of  the  lengths  of  straight  lines  to  a  certain  fixed 
line.  The  ratio  of  the  length  of  the  sum  of  the  lines  to  the  length 
of  the  fixed  lines  is  equal  to  the  sum  of  the  numbers.  The  con- 
struction also  applies  to  positive  and  negative  numbers,  if  we 
represent  them  by  the  ratio  of  the  length  of  straight  lines  of 
opposite  directions  to  the  length  of  a  fixed  line. 

In  order  to  multiply  a  given  quantity  c  by  a  given  number, 
let  the  number  be  given  as  the  ratio  of  the  lengths  of  two  straight 
lines  a/6.  If  the  quantity  c  is  also  represented  by  a  straight  line, 
all  we  have  to  do  is  to  find  a  straight  line  x  whose  length  is  to 
the  length  of  c  as  a  to  b.  This  can  be  done  in  many  ways  by 


FIG.  3. 


FIG.  4. 


constructing  any  triangle  with  two  sides  equal  to  a  and  6  and 
drawing  a  similar  triangle  with  the  side  that  corresponds  to  b  made 
equal  to  c.  As  a  rule  it  is  convenient  to  draw  a  and  b  at  right 
angles  and  the  similar  triangle  either  with  its  hypotenuse  parallel 
(Fig.  3)  or  at  right  angles  (Fig.  4)  to  the  hypotenuse  of  the  first 
triangle.  Division  by  a  given  number  is  effected  by  the  same  con- 
struction; for  the  multiplication  by  the  ratio  a/b  is  equivalent 
to  the  divisions  by  the  ratio  b/a. 

If  a,  b,  c  are  any  given  numbers,  we  can  represent  them  by  the 
ratios  of  three  straight  lines  to  a  fixed  line.     Then  the  ratio  of 


GRAPHICAL  CALCULATION. 


the  line  constructed  in  the  way  shown  in  Fig.  3  and  Fig.  4  to 
the  fixed  line  is  equal  to  the  number 


ac 


Multiplication  and  division  are  in  this  way  carried  out  simul- 
taneously. In  order  to  have  multiplication  alone,  we  need  only 
make  b  equal  1  and  in  order  to  have  division  alone,  we  need  only 
make  a  or  c  equal  1. 

In  order  to  include  the  multiplication  and  division  of  positive 
and  negative  numbers  we  can  proceed  in  the  following  way.  Let 
the  lines  corresponding  to  a,  x,  Fig.  3,  be  drawn  to  the  right  side 
of  the  vertex  to  signify  positive  numbers  and  to  the  left  side  to 
signify  negative  numbers.  Similarly  let  the  lines  corresponding 
to  b,  c  be  drawn  upward  to  signify  positive  numbers  and  down- 
ward to  signify  negative  numbers.  Then  the  drawing  of  a 
parallel  to  the  hypotenuse  of  the  rectangular  triangle  a,  b  through 
the  end  of  the  line  corresponding  to  c  will  always  lead  to  the 
number 


whatever  the  signs  of  a,  b,  c  may  be. 

The  same  definition  will  not  hold  for  the  construction  of  Fig.  4. 
If  the  positive  direction  of  the  line  corresponding  to  a  is  to  the 
right  and  the  positive  direction  of  the  line  corresponding  to  b  is 
upwards  then  the  positive  directions  of  x  and  c  ought  to  be  such 
that  when  the  right-angled  triangle  x,  c  is  turned  through  an 
angle  of  90°  to  make  the  positive  direction  of  x  coincident 
with  the  positive  direction  of  a,  the  positive  direction  of  c  coin- 
cides with  the  positive  direction  of  b.  If  we  wish  to  have  the 
positive  direction  of  x  upward,  the  positive  direction  of  c  would 
have  to  be  to  the  left,  or  if  we  wish  to  have  the  positive  direction 
of  c  to  the  right,  the  positive  direction  of  x  would  have  to  be 
downward.  If  this  is  adhered  to,  the  construction  for  division 
and  multiplication  will  include  the  signs. 


6 


GRAPHICAL   METHODS. 


§  3.  Integral  Functions. — We  have  shown  how  to  add,  subtract, 
multiply,  divide  given  numbers  graphically  by  representing  them 
as  ratios  of  the  lengths  of  straight  lines  to  the  length  of  a  fixed 
line  and  finding  the  result  of  the  operation  as  the  ratio  of  the 
length  of  a  certain  line  to  the  same  fixed  line.  By  repeating 
these  constructions  we  are  now  enabled  to  find  the  value  of  any 
algebraical  expression  built  up  by  these  four  operations  in  any 
succession  and  repetition.  Let  us  see  for  instance  how  the  values 
of  an  integral  function  of  x,  that  is  to  say,  an  expression  of  the  form 

may  be  found  by  geometrical  construction,  where  a0,  Oi  •  •  •  an,  x 

are  any  positive  or  negative 
numbers.  We  shall  first  as- 
sume that  all  the  numbers  are 
positive,  but  there  is  not  the 
least  difficulty  in  extending 
the  method  to  the  more  gen- 
eral case. 

Now  let  a0,  ai,  02,  •  •  •  an 
signify  straight  lines  laid  off 
on  a  vertical  line  that  we  call 
the  y-axis,  one  after  the  other 
as  if  to  find  the  straight  line 


ifc<C 


FIG.  5. 

The  lengths  of  these  lines  measured  in  a  conveniently  chosen 
unit  of  length  are  equal  to  the  numbers  designated  by  the  same 
letters.  In  Fig.  5  a0  runs  from  the  point  0  to  point  C\,  ai  from 
Ci  to  Cz,  "•  an  from  Cn  to  C«+i. 

Let  *  be  the  ratio  of  the  lines  Ox  and  01,  Fig.  5,  drawn  hori- 
zontally from  0  to  the  right.  The  length  01  is  chosen  of  con- 
venient size  independent  of  the  unit  of  length  that  measures  the 
lines  o0,  ai,  •  •  •  an.  The  length  Ox  is  then  defined  by  the  value 


GRAPHICAL   CALCULATION. 


of  the  ratio  x.  Through  x  and  1  draw  lines  parallel  to  the  z/-axis. 
Through  Cn+i  draw  a  line  parallel  to  Ox,  that  intersects  those  two 
parallels  in  Pn  and  Bn.  Draw  the  line  BnCn  that  intersects  the 
parallel  through  x  in  Pn_i.  Then  the  height  of  Pn_i  above  Cn 
will  be  equal  to  anx.  For  if  we  draw  a  line  through  Pn_i  parallel 
to  Ox  intersecting  the  y-axis  in  Dn,  the  triangle  CnDnPn-i  will  be 
similar  to  CnCn+iBn  and  their  ratio  is  equal  to  x,  therefore 
CnDn  =  anx.  Consequently  the  height  of  Pn_i  above  Cn-i  is 
equal  to  Cn-\Dn  =  anx  +  fln-i-  Now  let  us  repeat  the  same 
operation  in  letting  the  point  Dn  take  the  part  of  Cn+i.  Through 
Dn  draw  a  line  parallel  to  Ox,  that  intersects  the  parallels  through 
x  and  1  in  Pn-i  and  Bn-i>  Draw  the  line  Bn-iCn-i  that  intersects 
the  parallel  through  x  in  Pn-2« 
Then  the  height  of  Pn_2  above 
Cn-i  will  be  equal  to 

and  the  height  above  Cn-z  will  be 
equal  to 


Continue  in  the  same  way.  Draw 
Pn_2.Bn_2  parallel  to  Ox,  draw 
Bn-zCn-z  and  find  the  point  Pn_3- 
Then  the  height  of  Pn_3  above  Cn_2  will  be 

(anx*  +  an. 
and  the  height  of  Pn-3  above  Cr, 


FIG.  6. 


Finally  a  point  P0  is  found  (see  Fig.  6  for  n  =  4)  by  the  inter- 
section of  BiCi  with  the  parallel  to  the  i/-axis  through  x,  whose 
height  above  0  is  equal  to 


anxn      an-i 


Let  us  designate  the  line  xPQ  by  y,  so  that 


8  GRAPHICAL  METHODS. 

y  =  anxn  +  an-iz"-1  + f  a&  +  a0, 

in  the  sense  that  y  is  a  vertical  line  of  the  same  direction  and 
length  as  the  sum  of  the  vertical  lines  anxn,  anr-ixn~1t  •  •  •  a\x,  OQ. 
The  same  construction  holds  good  for  values  of  x  greater  than 
1  or  negative.  The  only  difference  is  that  the  point  x  is  beyond 
the  interval  01  to  the  right  of  1  or  to  the  left  of  0.  The  negative 
sign  of 

anx,  anx  +  an_i,  anx^  +  a^ix,  etc., 


will  signify  that  the  direction  of  the  lines  is  downward.  Nor  are 
any  alterations  necessary  in  order  to  include  the  case  that  several 
or  all  of  the  lines  a0,  a\,  —  •  an  are  directed  downward  and  corre- 
spond to  negative  numbers.  They  are  laid  off  on  the  y-axis  in 
the  same  way  as  if  to  find  the  sum 

flo  +  fli  +  02  +  •  •  •  +  On, 

(?„+!  lying  above  or  below  Ca  according  to  aa  being  directed 
upward  or  downward.  The  construction  can  be  repeated  for  a 
number  of  values  of  x.  The  points  P0  will  then  represent  the 
curve,  whose  equation  is 

y  =  a0  +  dix  +  •  •  •  +  anxn, 

x  and  y  measuring  abscissa  and  ordinates  in  independent  units 
of  length. 

In  order  to  draw  the  curve  for  large  values  of  z  a  modification 
must  be  introduced.  It  will  not  do  to  choose  01  small  in  order 
to  keep  x  on  your  drawing  board;  for  then  the  lines  J5.C.  will 
become  too  short  and  thus  their  direction  will  be  badly  defined. 
The  way  to  proceed  is  to  change  the  variable.  Write  for  instance 
X  =  z/10,  so  that  X  is  ten  times  as  small  as  x  and  write 

Then  as 

y  Oil  JQ  jQ2  n  jQn 


GRAPHICAL  CALCULATION.  9 

we  find 

y  =  Ao  +  AiX  +  A2X2  +  .  .  .  +  AnX\ 

Lay  off  the  lines  A0,  A\t  •  •  •  ^4n  in  a  convenient  scale  and  let 
Z  play  the  part  that  x  played  before.  The  curve  differs  in  scale 
from  the  first  curve  and  the  reduction  of  scale  may  be  different 
for  abscissas  and  ordinates  but  may  if  we  choose  be  made  the 
same  so  that  it  is  geometrically  similar  to  the  first  curve  reduced 
to  one  tenth.  It  is  evident  that  any  other  reduction  can  be 
effected  in  the  same  manner.  By  increasing  the  ratio  x/X  we 
enhance  the  value  of  A  n  in  comparison  to  the  coefficients  of  lesser 
index,  so  that  for  the  figure  of  the  curve  drawn  in  a  very  small 
scale  all  the  terms  will  be  insignificant  except  AnXn.  In  this 
case  the  points  C\,  Cz,  •  •  •  ,  Cn  will  very  nearly  coincide  with  0 
and  only  Cn+i  will  stand  out. 

It  is  interesting  to  observe  that  the  best  way  of  calculating  an 
integral  function 


for  any  value  of  x  proceeds  on  exactly  the  same  lines  as  the 
geometrical  construction.  The  coefficient  an  is  first  multiplied 
with  x  and  an_i  is  added  Call  the  result  an_i'.  This  is  again 
multiplied  by  x  and  an_2  is  added.  Call  this  result  an-2'.  Con- 
tinuing in  this  way  we  finally  obtain  a  value  of  a0',  which  is  equal 
to  the  value  of  the  integral  function  for  the  value  of  x  considered. 
Using  a  slide  rule  all  the  multiplications  with  x  can  be  effected 
with  a  single  setting  of  the  instrument.  The  coefficients  aa  and 
the  values  a*  are  best  written  in  rows  in  this  way 

On       drtr-l      On-2        "  '    Ci\  O,Q 

anx     an-i'x  •  •  •  az'x    ai'x 


The  accuracy  of  the  slide  rule  is  very  nearly  the  same  as  the 
accuracy  of  a  good  drawing.  But  the  rapidity  is  very  much 
greater.  When  therefore  only  a  few  values  of  the  integral  func- 
tion are  required,  the  geometrical  construction  will  not  repay 


10 


GRAPHICAL  METHODS. 


the  trouble.  It  is  different,  however,  when  the  object  is  to  make 
a  drawing  of  the  curve.  The  values  supplied  by  calculation 
would  have  to  be  plotted,  while  the  geometrical  construction 
furnishes  the  points  of  the  curve  right  away  and  in  this  manner 
gains  on  the  numerical  method. 

There  is  another  geometrical  method,  which  in  some  cases 
may  be  just  as  good.  Let  us  propose  to  find  the  value  of  an 
integral  function  of  the  fourth  degree. 

y  =  a0  +  aix  +  ctzx2  +  asz3  +  0,43* 

and  let  all  coefficients  in  the  first  instance  be  positive. 
The  coefficients  ao,  a-i,  02,  as,  a4  are  supposed  to  be  represented 

by  straight  lines,  while  x  will  be  the  ratio  of  two  lines.     The  lines 

OD,  01,  02,  03,  o4  are  laid  off  in  a 
broken  line  ao  to  the  right  from 
Co  to  Ci,  ai  upward  from  Ci  to 
C2,  02  to  the  left  from  C2  to  C3, 03 
downward  from  C3  to  C4,  o4  again 
to  the  right  from  C4  to  C5  (Fig.  7). 
Through  C5  draw  a  line  C$A  to 
a  point  A  on  C3C4  or  its  prolonga- 
tion and  let  x  be  equal  to  the 
ratio  CiA  :  C4C6  taken  positive 
when  C±A  has  the  same  direc- 


-^- 

D 

^ 

J***     as 

> 

\ 

•£     \ 

"\,                   / 

% 

!fc\ 

\ 

\ 

jr  ^  / 

\  j 

C0               ao           £'          (a 

Fio.  7. 

tion  as  Cad.    Then  we  have 


and 


i     =  o4z, 
C3A  =  0^  + 


(7^4  and  Cs^l  are  positive  or  negative  according  to  their  direction, 
being  the  same  as  the  direction  of  C3(74  or  opposite  to  it.  Through 
A  draw  the  line  AB  forming  a  right  angle  with  C&A  to  a  point  B 
on  CzCz  or  its  prolongation.  Then  we  have 


and 


C3A 


o3)  x 


GRAPHICAL   CALCULATION. 


11 


CzB  =  aix*  +  azx  -f-  02. 

CSB  and  C2B  are  positive  or  negative  according  to  their  direction 
being  the  same  as  the  direction  of  C2Cs  or  opposite  to  it.  'Simi- 
larly we  get 

CiD  =  a^  +  asz2  +  022  +  ai, 
and  finally 

azy?  +  c^z2  +  fliz  +  OQ. 


CoE  is  positive,  when  E  is  on  the  right  side  of  Co  and  negative 

when  on  the  left  side.     When  the  point  A  moves  along  the  line 

CzCi,  the  point  E  will  move 

along   the  line   Cod  and  its 

position   will    determine    the 

values  of  the  integral  function. 

To  find  the  position  of  E  for 

any  position  of  A,  we  might 

use  transparent  squared  paper,     /u^ 

that  we  pin  onto  the  drawing 

at  C5,  so  that  it  can  freely  be 

turned  round  €5.     Following 

FIG.  8. 
the  lines  of  the  squared  paper 

along  C&ABDE  after  turning  it  through  a  small  angle  furnishes 
the  position  of  E  for  a  new  position  of  A  (Fig.  8). 

To  include  the  case  of  negative  coefficients  we  draw  the  corre- 
sponding line  in  the  opposite  direction.  If  for  instance  03  is 
negative  C3C4  would  have  to  lie  above  C3;  but  C3A  would  have 
to  be  counted  in  the  same  way  as  before,  positive  in  a  downward, 
negative  in  an  upward  direction. 

The  extension  of  the  method  to  integral  functions  of  any  degree 
is  obvious  and  need  not  be  insisted  on.  It  may  be  applied  with 
advantage  to  find  the  real  roots  of  an  equation  of  any  degree. 
For  this  purpose  the  broken  line  C^ABDE  would  have  to  be 
drawn  in  such  a  way  that  E  coincides  with  CQ.  In  the  case  of 
Fig.  7,  for  instance,  it  is  easily  seen  that  no  real  root  exists. 
Fig.  9  shows  the  application  to  the  quadratic  equation.  A  circle 


12 


GRAPHICAL  METHODS. 


is  drawn  over  CoC3  as  diameter.    Its  intersections  with 
furnish  the  points  A  and  A'  that  correspond  to  the  two  roots. 
Both  roots  are  negative  in  this  case. 

The  first  method  of  constructing 
the  values  of  an  integral  function  can 
be  extended  to  the  case  where  the 
function  is  given  as  the  sum  of  a 
number  of  polynomials  of  the  form 


y  =  aQ  +  ai(x  —  p)  +  a*(x—  p)  (x  —  q) 
+  as(x  -  p)(x  -  q)(x  -  r)  +  •  •  •. 

Let  us  again  suppose  a0,  ai,  02,  •  •  • 
to  represent  straight  lines  laid  off  as 
before  on  the  ?/-axis  upwards  or  down- 

wards as  if  to  find  their  sum.  x,p,q,r  •  •  •  are  meant,  to  be  num- 
bers represented  by  the  ratio  of  certain  segments  on  the  axis  of 
abscissas.  Let  us  consider  the  case  of  four  terms,  the  highest  poly- 
nomial being  of  the  third  degree.  The  fixed  distance  between  the 
points  marked  p  and  p  +  1,  q  and  q-\-  I,  r  and  r  -f-  1  on  the 
axis  of  abscissas,  Fig.  10  is  chosen  arbitrarily  and  the  position 


^ 

r 

\ 

^ 

Qo 

"• 

,'"'' 

1 

-G 

A- 

-::i^ 

"Q> 

1 

"'( 

°2 

\ 

p        2     r        x  JM-1      gfl  <rtl           o;^ 
FIG.  10. 

of  the  points  marked  p,  q,  r,  x  is  made  such  that  the  ratio  of 
Op,  Oq,  Or,  Ox  to  that  fixed  distance  is  equal  to  the  numbers 
p,  q,  r,  x.  For  negative  values  the  points  are  taken  on  the  left 
of  0. 


GRAPHICAL  CALCULATION.  13 

Draw  parallels  to  the  ?/-axis  through  p,  q,  r,  x,  p  +  1,  q  +  1, 
r  +  1.  On  the  parallel  through  r  +  1  find  the  point  Qo  of  the 
same  ordinate  as  C\  and  on  the  parallel  through  r  find  the  point 
AQ  of  the  same  ordinate  as  C&  Join  AQ  and  QQ  by  a  straight 
line  and  find  its  intersection  PI  or  that  of  its  prolongation  with 
the  parallel  through  x.  The  height  of  PI  above  C3  or  AQ  is 
equal  to  03(2  —  r)  and  the  height  above  Cz  is  equal  to  a3(x  —  r) 
+  02.  On  the  parallel  through  q  +  1  find  a  point  Qi  of  the  same 
ordinate  as  PI  and  on  the  parallel  through  q  a  point  A\  of  the 
same  ordinate  as  Cz.  Join  A\  and  Qi  by  a  straight  line  and  find 
its  intersection  P2  or  that  of  its  prolongation  with  the  parallel 
through  x.  The  height  of  PZ  above  Cz  or  A\  is  equal  to 

[az(x  —  r)  +  av](x  —  q), 
and  the  height  above  C\  is  equal  to 

a3(x  -  r)(x  -  q)  +  a*(x  -  q)  +  «i. 

Finally  find  a  point  Qz  on  the  parallel  through  p  +  1  of  the 
same  ordinate  as  P2  and  a  point  A2  on  the  parallel  through  p  of 
the  same  ordinate  as  C\.  Join  ^42  and  Qz  by  a  straight  line  and 
find  its  intersection  P$  or  that  of  its  prolongation  with  the  par- 
allel through  x.  The  height  of  P3  above  Ci  or  Az  will  then  be 
equal  to 

MX  -  r)(x  -  q)  +  az(x  -  q)  +  a^(x  -  p) 

and  the  ordinate  of  P3  will  be  equal  to  the  given  integral  function 

y  =  a3(x  -  r)(x  -  q)(x  -  p)  +  a*(x  -  q)(x  -  p) 

+  ai(z  —  p)  +  a0. 

For  large  numbers  p,  q,  r,  x  we  use  a  similar  device  as  before  by 
introducing  new  numbers  P,  Q,  R,  X  equal  to  one  tenth,  or  one 
hundredth  or  any  other  fraction  of  pqrx.  For  instance 

P  =  p/10,    Q  =  9/10,    R  =  r/10     X  =  r/10. 
We  then  write 

AQ  =  a0,    Ai  =  10ab    A2  =  10002,     A3  =  1000a3, 


14  GRAPHICAL  METHODS. 

and  obtain 

y  =  A,  +  A1(X  -P)  +  A2(X  -  P)(X  -  Q) 

+  A*(X  -  P)(X  -  Q)(X  -  R). 

The  scale  for  the  lines  A0,  A\,  Az,  A3  and  y  must  then  be  reduced 
conveniently  and  the  values  are  constructed  in  the  same  way  as 
before. 

Now  let  us  consider  the  inverse  problem.  The  values  of  the 
integral  function  are  given  for 

x  =  p,  q,  r,  s; 

find  the  lines  a0,  a\t  (h,  OB,  so  that  the  value  of  the  integral  function 
may  be  found  for  any  other  value  of  x  in  the  way  shown  above. 

Let  us  designate  the  given  values  of  the  integral  function  for 
x  =  p,  q,  r,  s  by  yp,  yq,  yr,  y9  and  the  points  on  the  parallels  through 
p,  q,  r,  s  with  these  ordinates  by  P,  Q,  R,  S  (see  Fig.  12). 

For  x  =  p  the  integral  function 

y  =  a0  +  ai(x  —  p)  +  02(2  —  p)(x  —  q}  +  az(x—  p)(x—q}(x—r} 

reduces  to  a0.  Therefore  we  have  yp  =  a0.  The  point  Ci  is 
found  by  drawing  a  parallel  to  the  axis  of  abscissas  through  P 

and  taking  its  intersection  with 
the  axis  of  ordinates. 
^-T|A  In  order  to  find   C2  draw  a 


straight  line  through  P  and  Q 


and  find  its  intersection  A  with 
~\B  the  parallel  through  p  +  1  (Fig. 

11).     A  parallel  to  the  axis  of 
abscissas  through  A    intersects 


$  the  axis  of  ordinates  in  C2.     For 

FlG  llm  the  differences  yq— yp  and  ya—  yp 

(writing  ya  for  the  ordinate  of 

A)  are  proportional  to  the  differences  of  the  abscissas  and  con- 
sequently in  the  ratio  (q  —  p)  :  1.     Therefore 

*-»-*£*-* 


GRAPHICAL  CALCULATION. 


15 


In  the  same  way  as  the  point  Q  on  the  parallel  through  q  we 
might  join  any  point  X  on  a  parallel  through  x  with  the  point  P, 
find  the  intersection  with  the  parallel  through  p  +  1  and  draw  a 
parallel  to  the  axis  of  abscissas.  The  point  of  intersection  of 


FIG.  12. 


this  parallel  with  the  vertical  through  x  let  us  call  X'  and  its 
ordinate  y'.     Then  we  have 


y  -  yp  = 


-  q)  +  as(x  -  q) (x  —  r). 


y-  yP^  a  , 

x-  p 

Let  us  carry  out  this  construction  not  only  for  x  =  q  but  also 
for  x  =  r  and  x  =  s.  This  leads  us  to  three  points  Q',  Rf,  S' 
on  the  verticals  through  q,  r,  s,  whose  ordinates  are  the  values 
of  the  integral  functions 


y'  =  (a0  + 


—  q)(x  —  r). 


In  this  way  we  have  reduced  our  problem.  Instead  of  having 
to  find  an  integral  function  of  the  third  degree  from  four  given 
points  P,  Q,  R,  S,  we  have  now  only  to  find  an  integral  function 
of  the  second  degree  from  three  given  points  Q',  R',  S'.  A  second 
reduction  is  effected  in  exactly  the  same  manner.  Q'  is  joined 
with  R'  and  S'  by  straight  lines  and  through  their  intersection 
with  the  vertical  through  q  +  1  parallels  to  the  axis  of  abscissas 
are  drawn  that  intersect  the  verticals  through  r  and  s  in  the 
points  R"  and  S"  respectively.  The  ordinates  of  these  points 
are  the  values  of  the  integral  function  y"  defined  by 


16  GRAPHICAL  METHODS. 


for  x  =  r  and  x  =  s,  or 

y"  =  ao  +  a\  +  02  +  a3(z  —  «")• 

The  horizontal  through  R"  intersects  the  axis  of  ordinates  in  the 
point  C3.  Finally  we  find  C4  by  drawing  a  parallel  to  the  axis 
of  abscissas  through  the  intersection  of  R"S"  or  its  prolongation 
with  the  vertical  through  r  +  1. 

Having  found  the  points  CiCzCsd  we  can  now  for  any  value 
of  x  construct  the  ordinate 

y  =  a0  +  ai(x  —  p)  +  02(2  —  p)(x  —  q) 

+  a3(x-  p)(x-  q)(x-  r), 

and  thus  draw  the  parabola  of  the  third  degree  passing  through 
the  four  points  P,  Q,  R,  S. 

The  construction  may  be  somewhat  simplified  first  by  making 
p  +  1  =  q.  Our  data  are  the  points  P,  Q,  R,  S,  and  we  are 
perfectly  at  liberty  to  make  the  vertical  through  p  +  1  coincide 
with  the  vertical  through  Q.  In  this  case  the  point  Q'  will 
coincide  with  Q.  The  parabola  of  the  second  degree  through  the 
points  Q'R'S'  is  again  independent  of  the  distance  between  the 
verticals  through  q  and  9+1  and  at  the  same  time  independent 
of  the  point  P.  Therefore  we  are  perfectly  at  liberty,  for  the 
construction  of  any  point  of  this  parabola,  to  make  the  vertical 
through  q  +  1  coincide  with  the  vertical  through  R  even  if  the 
distance  of  the  verticals  through  P  and  Q  is  different  from  that 
of  the  verticals  through  Q  and  R.  R"  will  in  this  case  coincide 
with  R'.  The  procedure  is  shown  in  Fig.  12.  Starting  from 
the  points  P,  Q,  R,  S  the  first  step  is  to  find  R',  S'  by  connecting 
R  and  S  with  P  and  drawing  horizontals  through  the  inter- 
sections Ar,  A,  with  the  vertical  through  q.  The  next  step  is  to 
find  S"  by  connecting  Q  (identical  with  Q')  with  S'  and  drawing 
a  horizontal  through  the  intersection  with  the  vertical  through  r. 
Now  the  straight  line  R"S"  can  be  drawn  (R"  being  identical 


GRAPHICAL   CALCULATION, 


17 


with  R').  On  the  vertical  through  any  point  x  take  the  inter- 
section with  R"S"  and  pass  horizontally  to  the  point  Ax'  on  the 
vertical  through  r.  Draw  the  line  Q'AX'  and  find  its  intersection 
with  the  vertical  through  x.  This  point  is  on  the  parabola 
through  Q'R'S'.  Pass  horizontally  to  the  point  Ax  on  the 
vertical  through  q  and  draw  the  line  AXP.  Its  intersection  with 
the  vertical  through  #  is  a  point  on  the  parabola  of  the  third 
degree  through  P,  Q,  R,  S. 

The  method  is  evidently  applicable  to  any  number  of  given 
points,  the  degree  of  the  parabola  being  one  unit  less  than  the 
number  of  points. 

The  methods  for  the  construction  of  the  values  of  an  integral 
function  may  be  applied  to  find  the  value  of  any  rational  function 

V  =  R(x). 

For  a  rational  function  can  always  be  reduced  to  the  form  of  a 
quotient  of  two  integral  functions 

R(x)  =  g 


Xlf 


Now  after  having  constructed  curves  whose  ordinates  give  the 

values  of  g\(x)  and  gz(x)  for  any  abscissa  x  (Fig.  13),  R(x)  is  found 

in  the  following  manner. 

Through  a  point  P  on  the 

axis  of  abscissas  draw  a 

parallel  to  the  axis  of  or- 

dinates.   Let   GI  and  G2 

be  the  points  whose  ordi- 

nates  are  equal  to  gi(x) 

and  gz(x).     Pass  horizon- 

tally from  GI  to  GI  on  the 

vertical    through    P    and 

from  Gz  to  Gz   on  the  axis  of  ordinates.    Draw  a  line  through 

P  and  Gz   and  produce  it  as  far  as  A  where  it  intersects  the 

horizontal  through  GI.     Then  R(x)  is  equal  to  the  ratio  G\A 

to  PO.    Gi'A  may  then  be  set  off  as  ordinate  on  the  vertical 


Fio.  13, 


18 


GRAPHICAL  METHODS. 


through  x  and  defines  the  point  M  whose  ordinate  is  equal  to 
R(x}  in  length,  when  OP  is  chosen  as  the  unit  of  length. 

§4.  Linear  Functions  of  Any  Number  of  Variables.  —  Let  us 
consider  a  linear  function  of  a  number  of  variables  xi,  x%  •  •  •  xn, 


aQ  + 


04X2  + 


anxn, 


where  a<»  ai,  a%>  •  •  •  o,n  are  given  numbers  positive  or  negative. 
The  question  is  how  the  value  of  this  linear  function  may  be 
conveniently  constructed  for  various  systems  x\,  Xz,  •  •  •  xn. 
Suppose  ao,  a\,  •  •  •  an  to  represent  horizontal  lines  directed  to 
the  right  or  left  according  to  the  sign  of  the  corresponding  number 
and  to  be  laid  off  on  an  horizontal  axis  in  succession  as  if  to  find 
the  sum 

ao  +  ai  +  03  +  •  •  •  +  an, 

aQ  begins  at  0  and  runs  to  C\,  Oj  begins  at  C\  and  runs  to  (72  and 
so  on  (Fig.  14).  The  numbers  xi,  x^,  •  •  •  xn  let  us  represent 


a8 


FIG.  14. 

by  ratios  of  lengths.  We  draw  a  vertical  line  through  0  and 
choose  a  point  P  on  the  horizontal  axis.  Then  let  xi  be  equal 
to  the  ratio  OlfPO,  x^  =  02/PO,  etc.  If  P  is  chosen  on  the  left 
of  0,  we  take  the  point  1  above  0  for  a  positive  value  of  xi  and 
below  0  for  a  negative  one  and  the  same  for  the  other  points. 
Mark  a  point  0  above  0  in  the  same  distance  from  0  as  P.  Join 
the  point  P  with  the  points  0,  1,  2,  3,  4,  •  •  •  and  draw  a  broken 


GRAPHICAL  CALCULATION. 


19 


line  OA§A\A<iAzA\  in  such  a  manner  that  AQ  is  on  the  vertical 
through  Ci  and  OA  Q  is  parallel  to  PO,  A\  on  the  vertical  through 
<72  and  A0Ai  parallel  to  PI,  A2  on  the  vertical  through  Cs  and  A\A^ 
parallel  to  P2  and  so  on.  Then  the  ordinate  y0  of  A0  will  have 
the  same  length  as  a0  and  will  be  directed  upward  when  the 
direction  of  «o  is  to  the  right,  and  downward  when  the  direction 
of  do  is  to  the  left.  The  difference  y\  —  yo  of  the  ordinates  of  A\ 
and  AQ  is  equal  in  length  to  CL\XI,  as  y\  —  yo  and  ai  have  the  same 
ratio  as  01  and  PO.  AI  will  be  above  or  below  AQ  according  to 
the  line  a\x\  being  directed  to  the  right  or  to  the  left  and  it  is 
understood  that  a\x\  has  the  same  direction  as  a\  for  positive 


FIG.  15. 

values  of  x\  and  a  direction  opposite  to  ai  for  negative  values 
of  a?i.  Thus  the  ordinate  y\  has  the  same  length  as  the  line 
ffo  4~  d\x\  and  its  direction  is  upward  or  downward  according  to 
the  direction  of  the  line  o0  +  «i^i  being  to  the  right  or  to  the  left. 
In  the  same  way  it  is  shown  that  the  ordinate  yz  of  the  point  A2 
is  equal  in  length  to 

do  +  CLiXi  +  02^2, 

and  2/3  to 

flfl  +  Ol^l  +  02^2  +  «3^3 

and  so  on,  the  direction  upward  or  downward  corresponding 
to  the  positive  or  negative  value  of  the  linear  function. 
If  the  values  of  x\,  x%,  •  •  •  xn  satisfy  the  equation 

a0  +  a&i  +  02^2  +  •  •  •  +  dnxn  =  0 
the  ordinate  yn  must  vanish,  that  is  to  say,  the  point  An  must 


20 


GRAPHICAL  METHODS. 


coincide  with  Cn+i,  the  end  of  the  line  an.  And  vice  versa  if  An 
and  Cn+i  coincide  the  equation  is  satisfied.  Consequently  if  we 
know  all  the  values  but  one  of  the  numbers  Xi,  Q,  •  •  •  xn  the 
unknown  value  can  be  found  graphically.  For  suppose  x3  to  be 


PIG.  16. 

the  unknown  value  we  can,  beginning  from  0,  find  the  broken 
line  as  far  as  A2  and  beginning  from  the  other  end  An  we 
can  find  it  as  far  as  A$  (Fig.  15).  A  parallel  to  AZA3  through  P 
furnishes  the  point  3  on  the  axis  of  ordinates.  If  Xi,  Xz,  •  •  •  Zn-i 
are  known  and  only  xn  not,  we  can  draw  the  broken  line  as  far 
as  An-i  and  as  An  has  to  coincide  with  Cn+i,  we  can  draw  a  parallel 
to  An-iAn  through  P  and  find  the  point  n  on  the  axis  of  ordinates 


FIG.  17. 

that  determines  the  value  xn  by  the  ratio  On/PO  or  OnfOo.  In 
Figs.  15  and  16  all  the  coefficients  ao,  ai,  •••,  are  positive.  A 
negative  coefficient  05  is  shown  in  Fig.  17.  The  only  difference 
is  that  Ce  lies  to  the  left  of  C&  and  consequently  the  broken  line 
passes  from  AI  back  to  A& 

If  we  keep  the  points  0,  1,  2,  •  •  •,  in  their  positions  but  change 
the  position  of  P  to  P'  (Fig.  18)  and  repeat  the  construction  of 


GRAPHICAL  CALCULATION. 


21 


the  broken  line,  we  obtain  OAJAiAj  •  •  •  instead  of  OAoAiAz 
The  ordinate  y*  of  the  point  Aar  is  evidently 

00    .        01  Oa 


and  therefore 


PO 


That  is  to  say,  by  changing  the  position  of  P  without  changing 
the  position  of  the  points  0,  1,  2,  •  •  •  we  can  change  the  scale  of 
the  ordinates  of  the  broken  line.  They  change  inversely  pro- 


o  (7i 


FIG.  18. 


portional  to  PO.  It  may  be  convenient  to  make  use  of  this 
device  in  order  to  make  the  ordinates  a  convenient  size  inde- 
pendent of  the  scale  that  we  have  chosen  for  the  points  0,  1,  2,  ••• 
that  determine  the  values 

_  01  _02 

xi  -  0Q,    a*  -  00,  •••• 

A  linear  equation  with  only  one  unknown  quantity 
GO  +  a&i  =  0 

is  solved  by  drawing  a  parallel  to  A0Ai  through  P.  Let  a  second 
equation  be  given  with  two  unknown  quantities 

xz  =  0. 


The  lines  bo,  bi,  62  are  laid  off  as  before.  Knowing  xi  as  the 
solution  of  the  first  equation  we  can  construct  the  broken  line 
corresponding  to  the  second  equation  and  as  B2  must 


22  GRAPHICAL  METHODS. 

coincide  with  the  end  of  bz,  we  can  draw  a  parallel  to 
through  P  and  find  Xz.    In  a  similar  manner  we  can  find  z3 
from  a  third  equation 

Co  +  CiZi  +  CzXz  +  CSX3  =  0, 

and  so  we  can  find  any  number  of  unknown  quantities,  if 
each  equation  contains  one  unknown  quantity  more  than  those 
before. 

In  the  general  case  when  n  unknown  quantities  are  to  be 
determined  from  n  linear  equations  each  equation  will  contain 
all  the  unknown  quantities,  and  therefore  we  cannot  find  them 
one  after  the  other  as  in  the  case  just  treated.  But  it  can  be 
shown  that  by  means  of  very  simple  constructions  the  general  case 
is  reduced  to  a  set  of  equations,  such  as  has  just  been  treated. 

A         A  A  -Let  us  begin  with  two 

equations   and  two  un- 


& 

V'V 


known  quantities. 


FlG-19- 

The  lines  aQ,  a\,  az  are  laid  off  on  a  horizontal  line  OA0AiAz  and 
the  lines  60,  61,  62  on  another  horizontal  line  0'B0BiB2  (Fig.  19). 
Now  let  us  join  0  and  0',  A0  and  B0,  A\  and  BI,  A2  and  B2  by 
straight  lines  and  let  us  draw  a  third  horizontal  line  intersecting 
them  in  the  points  0"C0CiC2.  These  points  correspond  to  a 
certain  linear  function 

Co  +  CiZi  +  CzZij, 

and  it  can  be  shown  that  it  vanishes  when  xi  and  Xz  are  the  same 
values  for  which  the  first  two  linear  functions  vanish.  Let  the 
distance  of  the  first  two  horizontal  lines  be  I  and  the  distance  of 
the  third  from  the  first  and  second  h  and  k.  Then  it  can  readily 
be  seen  that 

Co  =  a0  +  -   (60  -  o0)  =      aQ  +  -    60. 


GRAPHICAL  CALCULATION.  23 

For  a  parallel  to  00'  through  AQ  defines  with  the  line  A0Bo  on 
the  third  and  second  horizontal  line  segments  equal  to  Co  —  flo 
and  60  —  OQ  and  as  these  segments  have  the  ratio  h/l,  it  follows 
that 

,   h  k  h 

CQ  =  aQ  +  y  (00  —  o0)  =  y  OD  +  y  DO- 

By  drawing  a  parallel  to  A0BQ  through  A\  and  to  A\B\  through 
Az  or  through  52  (which  comes  to  the  same  thing),  we  convince 
ourselves  in  the  same  way  that 

Ci  =  oi  +  y  (&i  —  01)  =  y  ai  +  y  k 
and 

02  =  02+  y  $2  —  02)  =  y  02  +  y  62. 

Multiplying  the  equation 

a0  +  OiZi  +  02^2  =  0 
by  fc/7  and  the  equation 

bo  +  to  +  ban  =  0 
by  A//  and  adding  the  two  products,  we  obtain 

C0  +  Wi  +  £2X2  =   0. 

The  third  horizontal  need  not  lie  between  the  first  two.  If  it 
lies  below  the  second  we  have  merely  to  give  k  a  negative  value 
and  if  it  lies  above  the  first  we  have  to  give  h  a  negative  value 
and  the  same  formulae  for  <?0,  Ci,  GZ  hold  good.  Consequently  the 
conclusion  remains  valid,  that  from  the  first  two  equations  the 
third  follows. 

Now  as  we  are  perfectly  at  liberty  to  draw  the  third  horizontal 
line  where  we  please,  we  can  let  it  run  through  the  intersection 
of  the  straight  lines  A\B\  and  A^Bz.  In  this  case  the  points  C\ 
and  Cz  must  coincide  and  consequently  c^  must  vanish.  If  c\ 
does  not  vanish  we  can  by  what  has  been  shown  above  find  .TI 
and  with  x\  we  can  find  a^  from  either  of  the  two  first  horizontal 


24  GKAPHICAL  METHODS. 

lines.  In  case  c\  also  vanishes,  that  is  to  say,  in  case  the  three 
straight  lines  A^Bz,  A\B\,  A^Bo  all  pass  through  the  same  point, 
while  00'  does  not  pass  through  it,  the  two  given  equations 
cannot  simultaneously  be  satisfied.  For  if  they  were,  it  would 
follow  that 

C0  +  CiXi  +  CzXz  =  0, 

and  as  c\  and  02  are  zero  c0  would  have  to  be  zero,  which  it  is  not 
as  00'  is  supposed  not  to  pass  through  the  intersection  of  AZBZ, 
A\B\  and  AQBo.  If  on  the  other  hand  all  four  lines  A2Bz,  A\B\, 
AoBo,  00'  pass  through  the  same  point,  CQ,  c\  and  c%  will  all  three 
vanish.  In  this  case  the  two  given  equations  do  not  contradict 
one  another,  but  &o&i&2  will  be  proportional  to  aoaiG^.  The 


Ai         AI      Aa         At  A,          At 


second  equation  will  therefore  contain  the  same  relation  between 
xi  and  xz  as  the  first,  so  that  there  is  only  one  condition  for  Xi 
and  xz  to  be  satisfied.  We  may  then  assign  any  arbitrary  value 
to  one  of  them  and  determine  the  value  of  the  other  to  satisfy  the 
equation. 

In  the  case  of  two  linear  equations  of  any  number  of  quantities 
%i)  Xz>  •  •  •  xnwe  can  by  the  same  graphical  method  eliminate  one 
of  the  quantities.  In  Fig.  20  this  is  shown  for  two  linear  equa- 
tions with  six  unknown  quantities.  The  two  horizontal  lines 
OAoAiAzAsAiAsAe  and  0'BQBiB2B3B^B5B6  represent  two  linear 
equations.  Through  the  intersection  of  A3B3  and  A4B4  a  third 
horizontal  line  is  drawn  intersecting  the  lines  00',  AoB0,  A\B\, 
- '  -  AfiBs  in  0"CQCi  •••  C6.  As  C3  and  (74  coincide,  the  line  c4 
vanishes  and  a-4  is  eliminated,  so  that  the  equation  assumes  the 
form 

Co  +  G\XI  +  fyXz+CsXt  -f-  c&a*5  +  CeZe  =  0- 


GRAPHICAL  CALCULATION.  25 

Suppose  now  that  a  set  of  six  equations  with  six  unknown  quan- 
tities is  represented  geometrically  on  six  horizontal  lines.  We  shall 
keep  one  of  these;  but  instead  of  the  other  five  we  construct  five 
new  ones  from  which  one  of  the  unknown  quantities  has  been 
eliminated  by  means  of  the  first  equation.  Now  it  may  happen 
that  at  the  same  time  another  unknown  quantity  is  eliminated, 
then  this  quantity  remains  arbitrary.  Of  the  five  new  equations 
we  again  keep  one  that  contains  another  unknown  quantity  and 
replace  the  four  others  again  by  four  new  ones  from  which  this 
unknown  quantity  has  been  eliminated.  Going  on  in  this 
manner  the  general  rule  will  be  that  with  each  step  only  one 
quantity  is  eliminated,  so  that  at  last  one  equation  with  one  un- 
known quantity  remains.  Instead  of  the  given  six  equations 
with  six  unknown  quantities  each,  we  now  have  one  with  six, 
one  with  five  and  so  on  down  to  one  with  one.  The  geometrical 
construction  shows  that  this  system  is  equivalent  to  the  given 
system,  for  we  can  just  as  well  pass  back  again  to  the  given 
system.  We  have  seen  above  how  the  unknown  quantities 
may  now  be  found  geometrically.  It  may  however  happen  in 
special  cases  that  with  the  elimination  of  one  unknown  quantity 
another  is  eliminated  at  the  same  time.  To  this  we  may  then 
assign  an  arbitrary  value  without  interfering  with  the  possibility 
of  the  solution.  Finally  all  unknown  quantities  may  be  elimi- 
nated from  an  equation.  If  in  this  case  there  remains  a  term 
different  from  zero  it  shows  that  it  is  impossible  to  satisfy  the 
given  equations  simultaneously.  If  no  term  remains,  the  two 
equations  from  which  the  elimination  takes  its  origin  contain  the 
same  relation  between  the  unknown  quantities  and  one  of  them 
may  be  ignored. 

§  5.  The  Graphical  Handling  of  Complex  Numbers.— A.  complex 
number 

z  =  x  +  yi 

is  represented  graphically  by  a  point  Z  whose  rectangular  coordi- 
nates correspond  to  the  numbers  x  and  y.     The  units  by  which 


26 


GRAPHICAL  METHODS. 


the  coordinates  are  measured,  we  assume  to  be  of  equal  length. 
We  might  also  say  that  a  complex  number  is  nothing  but  an 
algebraical  form  of  writing  down  the  coordinates  of  a  point  in  a 
plane.  And  the  calculations  with  complex  numbers  stand  for 
certain  geometrical  operations  with  the  points  which  correspond 
to  them. 

By  the  "sum"  of  two  complex  numbers 

21  =  xi  +  yii    and    22  =  a&  +  y*i 
we  understand  the  complex  number 


where 

and  we  write 


23 


and    y3 

=  2i  +  22. 


Graphically  we  obtain  the  point  Zz  representing  23  from  the 
points  Zi  and  Z2  representing  zi  and  22  by  drawing  a  parallel 
to  OZZ  through  Z\  and  making  Z\P 
(Fig.  21)  equal  to  OZ2  in  length 
and  direction  or  by  drawing  a  paral- 
lel through  Z2  and  making  Z2P 
equal  to  OZi  in  length  and  direc- 
tion. The  coordinates  of  P  are 
evidently  equal  to  #1  +  Xz  and 
2/i  +  2/2. 

Two  complex  numbers  2  and  z' 
are  called  opposite,  when  their  sum 
is  zero. 


FIG.  21. 


z  +  z'  =  0    or    x  =  —  x'    and 


or    2  =  —  z 


y-  -  y 

The  corresponding  points  Z  and  Z'  are  at  the  same  distance  from 
the  origin  0  but  in  opposite  directions. 

The   difference   of   two   complex   numbers   is   that   complex 
number,  which  added  to  the  subtrahend  gives  the  minuend 

22  +   (2l  —  %)   =  2l- 


GRAPHICAL  CALCULATION. 


27 


Therefore 


21  —  %  =  (zi  —  a*)  +  (1/1  — 


This  may  also  be  written 

zi  +  22'    where    22'=  —  22  =  —  a^  —  2/2*. 

That  is  to  say,  the  subtraction  of  the  complex  number  22  from  zi 
may  be  effected  by  adding  the  opposite  number  —  22.  For  the 
geometrical  construction  of  the  point  Z  corresponding  to  z\  —  & 
we  have  to  draw  a  parallel  to  OZ2  through  Z\  and  from  Z\  in 
the  direction  from  Z2  to  0  we  have  to  lay  off  the  distance  ZZ0. 
Or  we  may  also  draw  from  0  a  line  equal  in  direction  and  in  length 
to  ZzZi.  This  will  also  lead  to  the  point  Z  representing  the 
difference  21  —  22. 

The  rules  for  multiplication  and  division  of  complex  numbers 
are  best  stated  by  introducing  polar  coordinates.  Let  r  be  the 
positive  number  measuring  the  distance  OZ  in  the  same  unit 
of  length  in  which  x  and  y  measure  the  abscissa  and  ordinate,  so 
that 


and  let  <p  be  the  angle  between  OZ  and  the  axis  of  x,  counted  in 
the  direction  from  the  positive  axis  of  x  toward  the  positive 
axis  of  y  through  the  entire 
circumference  (Fig.  22).  Then 
we  have 

x  =  r  cos  <p,    y  =  r  sin  <p 
and 
2  =  x  +  yi  =  r(cos<f>  +  sin  (pi). 

Let  us  call  r  the  modulus 
and  tp  the  angle  of  z.  The  an- 
gle  may  be  increased  or  di- 

minished by  any  multiple  of  four  right  angles  without  altering 
z,  but  any   alteration  of  r  necessarily  implies  an  alteration  of  z. 


22. 


28  GRAPHICAL  METHODS. 

According  to  Moivre's  theorem,  we  can  write 

z  =  re**'. 
By  the  product  of  two  complex  numbers 

zi  =  rie*"    and    22  =  tfce*** 

we  understand  that  complex  number  23  whose  modulus  r$  is 
equal  to  the  product  of  the  moduli  r\  and  r2  and  whose  angle  ^ 
is  the  sum  of  the  angles  <p\  and  <&  or  differs  from  the  sum  only  by 
a  multiple  of  four  right  angles 


The  definition  of  division  follows  from  that  of  multiplica- 
tion. The  quotient  Zi  divided  by  22  is  that  complex  number, 
which  multiplied  by  22  gives  Zi.  Therefore  the  product  of  its 
modulus  with  the  modulus  of  Za  must  be  equal  to  the  modulus  of 
Zi  and  the  sum  of  its  angle  with  the  angle  of  22  must  be  equal  to 
the  angle  of  z\.  Or  we  may  also  say  the  modulus  of  the  quotient 
Zi/Z2  is  equal  to  the  quotient  of  the  moduli  TI/TZ  and  its  angle  is 
equal  to  the  difference  of  the  angles  <p\  —  <&•  An  addition  or 
subtraction  of  a  multiple  of  four  right  angles  we  shall  leave  out 
of  consideration  as  it  does  not  affect  the  complex  number  nor 
the  point  representing  it. 

The  geometrical  construction  corresponding  to  the  multi- 
plication and  division  of  complex  numbers  is  best  described  by 
considering  two  quotients  each  of  two  complex  numbers  that 
give  the  same  result.  Let  us  write 

2l/Z2  =  Z3/Z4. 

The  geometrical  meaning  of  this  is  that 

rifa  =  r3/r4, 
and 

<f>i  —  <f>z  —  <f>3  —  <p\. 


That  is  to  say,  the  triangles  Z^OZz  and  Z3OZ4  are  geometrically 


GRAPHICAL   CALCULATION. 


29 


similar  (Fig.  23).  When  three  of  the  points  Zlt  Z2,  Z3,  Z4  are 
given  the  fourth  can  evidently  be  found.  For  instance  let 
Z\,  Zz,  Z\  be  given.  Draw  a  parallel  to  Z\Z^.  intersecting  OZ% 
at  a  distance  r4  from  0.  This  point  together  with  the  inter- 
section on  OZi  and  with  0  will  form  the  three  corners  of  a  tri- 
angle congruent  to  the  triangle  Z4Z30.  It  will  be  brought  into 


FIG.  23. 


FIG.  24. 


the  position  of  Z4Z30  by  being  turned  round  0  so  as  to  bring  the 
direction  of  the  side  in  OZ2  into  the  position  of  0Z4.  Thus  the 
direction  of  OZ3  and  its  length  may  be  found. 

This  construction  contains  multiplication  as  well  as  division  as 
special  cases.  Let  Z4  coincide  with  the  point  x  =  1,  y  =  0,  so 
that  z4  =  1  (Fig.  24),  then  we  have 

Zl/Z2  =  3j      OF      Zi  =  22Z3. 

From  any  two  of  the  points  Z\,  Z2,  Z3  a  simple  construction  gives 
us  the  third. 

The  geometrical  representation  of  complex  numbers  may  be  used 
to  advantage  to  show  the  properties  of  harmonic  oscillations. 

Let  a  point  P  move  on  the  axis  of  x,  so  that  its  abscissa  at  the 
time  t  is  given  by  the  formula 

x  =  r  cos  (nt  +  a), 
n,  r  and  a  being  constants.     We  call  r  the  amplitude  and  nt  +  a 


30  GRAPHICAL  METHODS. 

the  phase  of  the  motion.  The  point  P  moves  backwards  and 
forwards  between  the  limits  x  =  r  and  x  =  —  r.  The  time 
T  =  2ir/n  is  called  the  period  of  the  oscillation,  it  is  the  time  in 
which  one  complete  oscillation  backwards  and  forwards  is  per- 
formed. 

Now  instead  of  x  let  us  consider  the  complex  number 

z  =  r  cos(nt  +  a)  +  r  sin(nt  +  a)i 
or 

2  =  re(n*+a)% 

of  which  x  is  the  abscissa  and  let  us  follow  the  movement  of 
the  point  Z.  For  t  =  0  we  have 

z  =  reai. 
Designating  this  value  by  z0,  we  can  write 

z  =  zoentt. 
The  geometrical  meaning  of  the  product 


is  that  the  line  OZ0  is  turned  round  0  through  the  angle  nt.  For 
the  modulus  of  enti  being  equal  to  1  the  modulus  of  z0  is  not 
changed  by  the  multiplication.  The 
movement  of  the  point  Z  therefore 
consists  in  a  uniform  revolution  of 
®Z  round  0.  At  the  moment  t=  0 
the  position  is  OZo  and  after  the 
time  T  =  2ir/n  the  same  position  is 


*=2/2' 


\  J 


occupied  again.  The  revolution  goes 
on  in  the  direction  from  the  positive 
axis  of  x  to  the  positive  axis  of  y 
(Fig.  25). 

FlQ  ^  The  movement  of  Z  is  evidently 

simpler  than  the  movement  of  the 
projection  P  of  Z  on  the  axis  of  x. 

Let  us  consider  a  motion  composed  of  the  sum  of  two  harmonic 


GRAPHICAL   CALCULATION.  31 

motions  of  the  same  period  but  of  different  amplitudes  and 
phases 

x  =  r  i  cos  (nt  +  en)  +  r2  cos  (nt  +  <*2), 

and  let  us  again  substitute  the  motion  of  the  point  Z  correspond- 
ing to  the  complex  number 

2  = 

For  t  =  0  the  first  term  is 
and  the  second  term 


Introducing  z\  and  %  into  the  expression  for  z  we  have 

z  =  Zlenti  +  sfee"*'  =  (21  +  z£enti  =  z3en« 
where 

23  =  2i  +  2z. 

This  shows  at  once  that  the  movement  of  Z  is  a  uniform  circular 
movement  consisting  in  a  uniform  revolution  of  OZ  round  0. 
The  position  at  the  moment  t  =  0  is  OZ$  corresponding  to  the 
complex  number 


The  projection  of  Z  on  the  axis  of  x  has  the  abscissa 

x  =  TZ  cos  (nt  +  «3) 

where  r3  and  a3  designate  modulus  and  angle  of  z3.  Thus  the 
sum  of  two  harmonic  motions  of  the  same  period  is  shown  also 
to  form  a  harmonic  motion. 

The  same  holds  for  a  sum  of  any  number  of  harmonic  motions 
of  the  same  period.     For  the  complex  number 


where  r\,  TZ,  •  •  •  r^;  ai,  02,  •  •  •  a*  and  n  are  constants  may  be 
written 

2  =  Zlenti  +  zzenti  +  •  •  • 
or 


32  GRAPHICAL  METHODS. 


z  =  2<>en"', 
where 

ZO  =  Zl  +  32  +    '  '  '   +  2A. 

The  movement  of  Z  therefore,  excepting  the  case  z0  =  0,  consists 
in  a  uniform  revolution  of  OZ  round  0,  OZ  always  keeping  the 
same  length  equal  to  the  modulus  of  z0.     The  position  of  OZ  at 
the  moment  t  =  0  is  OZ0. 
The  motion  of  a  point  P  whose  abscissa  is 

x  =  ae~kt  cos  (id  +  a) 

where  a,  k,  n,  a  are  constants  (a  and  k  positive)  is  called  a  damped 
harmonic  motion.  It  may  be  looked  upon  as  a  harmonic  motion, 
whose  amplitude  is  decreasing.  To  study  this  motion  let  us 
again  substitute  a  complex  number 

z  =  ae~kt  cos  (rd  +  a)  +  ae~kt  sin  (nt  +  a)i, 
or 

z  =  ae-kt'eW», 
or 

z  =  z0<r*<.e"ri, 

where  ZQ  is  written  for  the  complex  constant  ae**. 
The  product 


is  a  complex  number  corresponding  to  a  point  Z\  on  the  same 
radius  as  Z0,  coincident  with  Z0  at  the  moment  t  =  0  but  ap- 
proaching 0  in  a  geometrical  ratio  after  t  =  0.  In  unit  of  time 
the  distance  of  Zi  from  0  decreases  in  the  constant  ratio  e~k  :  1. 
The  multiplication  with  enti  turns  OZi  round  0  through  an  angle 
nt.  We  may  therefore  describe  the  motion  of  Z  as  a  uniform 
revolution  of  OZ  round  0,  Z  at  the  same  time  approaching  0 
at  a  rate  uniform  in  this  sense  that  in  equal  times  the  distance 
is  reduced  in  equal  proportions  (Fig.  26).  At  the  moment  t  =  0 
the  position  coincides  with  ZQ.  We  speak  of  a  period  of  this 
motion  meaning  the  time  T  =  2ir/n  in  which  OZ  performs  an 
entire  revolution  round  0,  although  it  does  not  come  back  to  its 


GRAPHICAL   CALCULATION. 


original  position.  Any  part  of  the  spiral  curve  described  by  Z 
in  a  given  time  is  geometrically  similar  to  any  other  part  of  the 
curve  described  in  an  interval  of  equal  duration.  For  suppose 
the  second  interval  of  time  hap- 
pens T  units  of  time  later,  we 
shall  have  for  the  first  interval 

z  =  z0e-kt-enti, 
and  for  the  second  interval 


Now  if  zi  and  Zz  are  the  values 
of  z  at  two  moments  ti  and  k  of 
the  first  interval  and  Zi  and  %' 
the  corresponding  values  of  z' 
at  the  moments  ti  +  T  and  ^  -f  T  of  the  second  interval,  we  have 


FIG.  26. 


22 


Therefore  the  triangle  ZiOZ%  is  geometrically  similar  to  the 
triangle  Zi'OZz'.  As  Z\  and  Zi  may  coincide  with  any  points 
of  the  first  part  of  the  curve,  the  two  parts  are  evidently  geo- 
metrically similar. 

The  projection  of  Z  on  the  axis  of  x  performs  oscillations 
decreasing  in  amplitude.  The  turning-points  correspond  to  those 
points  of  the  spiral  curve  described  by  Z,  where  its  tangent  is 
parallel  to  the  axis  of  y,  that  is  to  say,  where  the  abscissa  of  dz/dt 
vanishes. 

Now 

dz 


or 


dx 


34  GRAPHICAL  METHODS. 

where  p  and  X  are  the  modulus  and  angle  of  the  complex  number 
—  k  +  ni. 

Consequently,  if  we  represent  dz/dt  by  a  point  Z',  the  triangle 
Z'OZ  will  remain  geometrically  similar  to  itself.  The  turning 
points  of  the  damped  oscillations  correspond  to  the  moments 
when  OZr  is  directed  vertically  upward  or  downward  or  when  the 
angle  of  dz/dt  is  equal  to  r/2  or  37T/2.  The  angle  of  z  will  then 
be  7T/2  —  X  or  3-7T/2  —  X  plus  or  minus  any  multiple  of  2w.  As 
the  angle  of  z,  on  the  other  hand,  is  changing  in  time  according 
to  the  formula 

rt+a, 

we  find  the  moments  where  the  movement  turns  by  the  equation 

nt  +  a  =     Tr/2  -  X  +  2Nir, 
or 

ni  +  a  =  37T/2  -  X  +  2Nv, 

N  denoting  any  positive  or  negative  integral  number.  The  time 
between  two  consecutive  turnings  is  therefore  equal  to  ir/n,  that 
is,  equal  to  half  a  period.  All  the  points  Z  corresponding  to 
turning  points  lie  on  the  same  straight  line  through  the  origin  0 
forming  an  angle  3^/2  —  X  with  the  direction  of  the  positive  axis 
of  x.  The  amplitudes  of  the  consecutive  oscillations  therefore 
decrease  in  the  same  proportion  as  the  modulus  of  z,  that  is 
to  say,  in  half  a  period  in  the  ratio  e~~Z' 

Let  us  consider  the  vibrations  of  a  system  possessing  one 
degree  of  freedom  when  the  system  is  subjected  to  a  force  varying 
as  a  harmonic  function  of  the  time  and  let  us  limit  our  considera- 
tions to  positions  in  the  immediate  neighborhood  of  a  position 
of  stable  equilibrium.  If  the  quantity  x  determines  the  position 
of  the  system  the  oscillations  satisfy  a  differential  equation  of 
the  form 


where  m,  k,  n,  p,  F  are  positive  constants. 

1  See  for  instance  Rayleigh,  Theory  of  Sound,  Vol.  I,  chap.  Ill,  §  46. 


GRAPHICAL   CALCULATION.  35 

This  is  another  case  where  the  introduction  of  a  complex 
variable 

z  =  x  +  yi 

and  the  geometrical  representation  of  complex  numbers  helps  to 
form  the  solution  and  to  survey  the  variety  of  phenomena  that 
may  be  produced. 

In  order  to  introduce  z  let  us  simultaneously  consider  the 
differential  equation 


and  let  us  multiply  the  second  equation  by  i  and  add  it  to  the 
first.     We  then  have 


The  movement  of  the  point  Z  representing  the  complex  number 
z  then  serves  as  well  to  show  the  movement  corresponding  to  x. 
We  need  only  consider  the  projection  of  Z  on  the  axis  of  x. 

A  solution  of  the  differential  equation  may  be  obtained  by 
writing 

z  =  z0epti. 

Introducing  this  expression  for  z  and  cancelling  the  factor  epti 
we  have 

zo(-  mp2  +  kpi  +  n2)  =  F, 
or 

F 


z0  is  a  complex  constant,  that  may  be  represented  geometrically 
as  we  shall  see  later  on. 
This  solution 


is  not  general.     If  z'  denotes  any  other  solution  so  that 


36  GRAPHICAL  METHODS. 


we  find  by  subtracting  the  two  equations 


or  writing 

z'  —  z  =  u, 


The  general  solution  of  this  equation  is 
u  =  u^  +  u^t 

where  u\  and  ^  are  arbitrary  constants  and  \\  and  X2  are  the 
roots  of  the  equation  for  X 

mX2  +  k\  +  n2  =  0, 


If  P/4m2  is  greater  than  n2,  so  that  the  square  root  has  a  real 
value,  V/^2/4m2  —  nz  will  certainly  be  smaller  than  k/2m.  There- 
fore Xi  and  X2  will  both  be  negative  and  the  moduli  of  the  complex 
numbers  Ui^lt  and  u^*'  will  in  time  become  insignificant.  If, 
on  the  other  hand,  ]t?/4.m2  is  smaller  than  n2,  both  complex 
numbers  u\f^1*  and  y^e^  correspond  to  points  describing  spirals  that 
approach  the  origin,  as  we  have  seen  above,  in  a  constant  ratio 
for  equal  intervals  of  time.  Therefore  they  will  also  in  time 
become  insignificant. 

After  a  certain  lapse  of  time  the  expression 

z  =  Zaepti 

will  therefore  suffice  to  represent  the  solution. 

The  point  Z  moves  uniformly  in  a  circle  round  0  of  a  radius 
equal  to  the  modulus  of  z0,  completing  one  revolution  in  the 
period  27r/p,  the  period  of  the  force  acting  on  the  system.  The 


GRAPHICAL   CALCULATION.  37 

movement  of  the  projection  of  Z  on  the  axis  of  x  is  given  by 

x  =  r0  cos  (pt  +  a), 

where  TQ  is  the  modulus  and  a  the  angle  of  ZQ.  It  is  a  harmonic 
movement  with  the  same  period  as  that  of  the  force*  Fcospt, 
but  with  a  certain  difference  of  phase  and  a  certain  amplitude 
depending  on  the  values  of  F,  m,  k,  n,  p. 

It  is  important  to  study  this  relation  in  order  to  survey  the 
phenomena  that  may  be  produced.  For  this  purpose  the  geo- 
metrical representation  of  complex  numbers  readily  lends  itself. 

In  the  expression  for  ZQ 

F 


let  us  consider  the  denominator 

-  mp2  +  kpi  +  «2, 

and  let  us  suppose  the  period  of  the  force  acting  on  the  system 
not  determined,  while  the  constants  of  the  system  m,  k,  n  and 
the  amplitude  of  the  force  F  have  given  values.  The  quantity  p 
is  the  number  of  oscillations  of  the  force  during  an  interval  of 
2ir  units  of  time.  This  quantity  p  we  suppose  to  be  indeter- 
minate and  we  intend  to  show,  how  the  amplitude  and  phase 
of  the  forced  vibrations  compare  with  the  amplitude  and  phase 
of  the  force  for  different  values  of  p. 

Let  us  plot  the  curve  of  the  points  corresponding  to  the  complex 
number 

n2  -  mp2  +  kpi, 

where  p  assumes  the  values  p  =  0  to  +00. 

This  curve  is  a  parabola  whose  axis  coincides  with  the  axis  of 
x  and  whose  vertex  is  in  the  point  x  =  n2,  y  =  0.  We  find  its 
equation  by  eliminating  p  from  the  equations 

x  =  n2  -  mpz,      y  =  kp, 
viz., 


38 


GRAPHICAL  METHODS. 


p-3 


But  it  is  better  not  to  eliminate  p  and  to  plot  the  different  points 

for  different  values  of  p.     In  Fig.  27  the  curve  is  drawn  for  p  =  0 

to  3  and  the  points  for 
p=Q,  1,  2,  3  are  marked. 
The  ordinates  increase  in 
proportion  to  p',  they  are 
equal  to  0,  k,  2k,  3k  for 
p  =  0,  1,  2,  3.  The  dis- 
tance between  the  projec- 
tion of  any  point  of  the 
curve  on  the  axis  of  x  and 

the  vertex  is  proportional  to  p2.    It  is  equal  to  0,  m,  4m,  9m  for 

p  =  0,  1,  2,  3. 

For  any  point  P  on  the  parabola  let  us  denote  the  distance 

from  0  by  r  and  the  angle  between  OP  and  the  positive  axis  of 

x  by  <p  so  that 

n2  —  mp2  +  kpi  =  re**. 

Then  we  have 

20  =  re*', 

and  consequently 


FIG.  27. 


2  =  - 


and 


=  -  cos  (pt  -  <f>). 


The  amplitude  F/r  of  the  forced  vibration  is  inversely  propor- 
tional to  r.  Thus  our  Fig.  27  shows  us  what  the  period  of  the 
force  must  be  to  make  the  forced  vibrations  as  large  as  possible. 
It  corresponds  to  the  point  on  the  parabola  whose  distance  from 
0  is  smallest.  It  is  the  point  where  a  circle  round  0  touches  the 
parabola.  In  Fig.  27  this  point  is  marked  R.  It  may  be  called 
the  point  of  maximum  resonance.  When  the  constants  of  the 
system  are  such  that  the  ordinate  of  the  point,  where  the  parabola 
intersects  the  axis  of  y  is  small  in  comparison  with  the  abscissa 


GRAPHICAL   CALCULATION.  39 

of  the  vertex,  then  OR  will  lie  close  to  the  axis  of  y  (Fig.  28).  In 
this  case  the  angle  between  OR  and  the  positive  axis  of  x  will  be 
very  nearly  equal  to  90°,  that  is  to  say,  the  forced  oscillations  will 
lag  behind  the  force  oscil- 
lations by  a  little  less  than  a 
quarter  of  a  period.  Keep- 
ing m  and  n  constant,  this 
will  take  place  for  small  val- 
ues of  k,  i.  e.,  for  a  small 
damping  influence.  A  small  FlG  2g 

deviation  of  p  from  the  fre- 
quency of  maximum  resonance  will  throw  the  point  P  awayfrom.R, 
so  that  r  increases  considerably  and  <p  becomes  either  very  small 
(for  values  of  p  smaller  than  the  frequency  of  maximum  resonance) 
or  nearly  equal  to  180°  (for  values  of  p  larger  than  the  frequency 
of  maximum  resonance).  In  other  words  for  small  values  of  k  the 
maximum  of  resonance  is  very  sharp.  A  deviation  of  the  period 
of  the  force  from  the  period  of  maximum  resonance  will  lessen  the 
amplitude  of  the  forced  vibration  considerably.  The  lag  of  its 
phase  behind  that  of  the  force  will  at  the  same  time  nearly  vanish, 
when  the  frequency  of  the  force  is  decreased  or  it  will  become  nearly 
as  large  as  half  a  period,  when  the  frequency  of  the  force  is  in- 
creased. For  larger  values  of  k  the  parabola  opens  out  and  this 
phenomenon  becomes  less  marked.  The  minimum  of  the  radius  r 
becomes  less  pronounced.  The  angle  between  OR  and  the  axis  of 
x  becomes  smaller  and  smaller  and  for  a  certain  value  of  k  and  all 
larger  values  the  point  R  will  coincide  with  the  vertex  of  the  para- 
bola. In  this  case,  there  is  no  resonance.  When  the  period  of 
the  force  increases  indefinitely  (p  becoming  smaller  and  smaller) 
the  amplitude  of  the  forced  vibration  will  increase  and  will 
approach  more  and  more  to  the  limit 

£ 

w2' 

but  there  will  be  no  definite  period  for  which  the  forced  vibra- 
tions are  stronger  than  for  all  others. 


CHAPTER  II. 

THE  GRAPHICAL  REPRESENTATION  OF  FUNCTIONS  OF  ONE  OR 
MOBE  INDEPENDENT  VARIABLES. 

§  6.  Functions  of  One  Independent  Variable. — A  function  y  of 
one  variable  x 

y  =  f(*) 

is  usually  represented  geometrically  by  a  curve,  in  such  a  way 
that  the  rectangular  coordinates  of  its  points  measured  in  certain 
chosen  units  of  length  are  equal  to  x  and  y.  This  graphical  rep- 
resentation of  a  function  is  exceedingly  valuable.  But  there  is 
another  way  not  less  valuable  for  certain  purposes,  more  used  in 
applied  than  in  theoretical  mathematics,  which  here  will  occupy 
our  attention. 

Suppose  the  values  of  y  are  calculated  for  certain  equidistant 
values  of  x,  for  instance: 

x  =  -  6,  -  5,  -  4,  -  3,   -2,  -  1,  0, 

+  1,  +  2,  +  3,  +  4,  +  5,  +  6, 

and  let  us  plot  these  values  of  y  in  a  uniform  scale  on  a  straight 

line.  Draw  the  uniform  scale  on 
one, side  of  the  straight  line  and 
mark  the  points  that  correspond 
to  the  calculated  values  of  y  on 
the  other  side  of  the  straight  line. 
Denote  them  by  the  numbers  x 
that  belong  to  them  (Fig.  29). 
The  drawing  will  then  allow  us  to 
read  off  the  value  of  y  for  any  of 
FlG  29  the  values  of  x  with  a  certain  ac- 

curacy depending  on  the  size  of  the 

scale  and  the  number  of  its  partitions  and  naturally  on  the  fine- 

40 


GRAPHICAL  REPRESENTATION   OF   FUNCTIONS.  41 

ness  of  the  drawing.  It  will  also  allow  us  to  read  off  the  value 
of  y  for  a  value  of  x  between  those  that  have  been  marked,  if 
the  intervals  between  two  consecutive  values  of  x  are  so  small 
that  the  corresponding  intervals  of  y  are  nearly  equal.  We  can 
with  a  certain  accuracy  interpolate  values  of  x  by  sight.  On  the 
other  hand,  we  can  also  read  off  the  values  of  x  for  any  of  the 
values  of  y.  We  shall  call  this  the  representation  of  a  function 
by  a  scale. 

We  can  easily  pass  over  to  the  representation  of  the  same 
function  by  a  curve.  We  need  only  draw  lines  perpendicular 
to  the  line  carrying  the  scales  through  the  points  marked  with 
the  values  of  x  and  make  their  length  measured  in  any  given 
unit  equal  to  the  numbers  x  that  correspond  to  them  (Fig.  29). 

In  a  similar  way  we  can  pass 
from  the  representation  of  the 
function  by  a  curve  to  the  rep- 
resentation by  a  scale. 

The  representation  by  a  scale 
may  be  imagined  to  signify  the 
movement  of  a  point  on  a  straight 
line,  the  values  of  x  meaning  the 
time  and  the  points  marked  with  ^ 
these  values  being  the  positions 
of  the  moving  point  at  the  times 

marked.  By  passing  over  to  the  curve  the  movement  in  the 
straight  line  is  drawn  out  into  a  curve  with  the  time  as  abscissa 
(Fig.  30).. 

The  representation  by  a  scale  is  used  in  connection  with  the 
representation  by  a  curve  for  the  purpose  of  drawing  a  function 
of  a  function. 

Let  y  be  a  function  of  x  and  x  a  function  of  t.  Then  we  wish 
to  represent  y  as  a  function  of  t. 

Let  y  =  f(x)  be  given  by  a  curve  in  the  usual  way  and  let 
x  =  <p(t)  be  given  by  a  scale  on  the  axis  of  x  marking  the  points 
where  t  =  0,  1,  2,  •  •  •,  12.  We  then  find  the  values  of  y  corre- 


42 


GRAPHICAL   METHODS. 


spending  to  the  values  t  =  0,  1,  2,  •  •  -,  12  by  drawing  the  ordi- 
nates  of  the  curve  y  =  f(x)  for  the  abscissas  marked  t  =  0,  1,  2, 


Fio.  31. 

•  •  •,  12.    These  ordinates  as  a  rule  will  not  be  equidistant.     But 

as  soon  as  we  move  them  so  as  to  make  them  equidistant,  they 

*  A    -PI       form  the  ordinates  of  the  curve 


\  \ 


y  = 


with  /  as  abscissa  (Fig.  31). 

The  representation  of  a  func- 
tion by  a  scale  may  be  general- 
ized in  the  respect  that  neither 
of  the  two  scales  facing  one  an- 
other on  the  straight  line  need 
necessarily  be  uniform.  The  in- 
tervals of  both  scales  may  vary 
from  one  side  of  the  scale  to  the 
other.  If  the  variation  is  suffi- 

ciently slow  the  interpolation  can  nevertheless  be  effected  with 
accuracy.  We  may  look  at  this  case  as  composed  of  two  cases 
of  the  first  kind. 

f(x)  =  y    and    y  =  g(t}. 


FIG.  32. 


GRAPHICAL   REPRESENTATION    OF   FUNCTIONS.  43 

These  scales  are  placed  together,  so  that  the  scale  x  touches  the 
scale  t 

/(*)  =  0(0, 

while  the  scale  y  is  cut  out  (Fig.  32). 

§  7.  The  Principle  of  the  Slide  Rule. — Let  us  investigate  how 
the  relation  between  x  and  t  changes  by  sliding  the  x-  and  ^-scales 
along  one  another. 

If  we  slide  the  z-scale  through  an  amount  y  =  c  so  that  a 
point  of  the  z-scale  that  was  opposite  to  a  certain  point  y  of  the 
?/-scale,  now  is  opposite  y  +  c,  then  the  relation  between  x  and  t 
represented  by  the  new  position  of  the  scales  will  be  given  by 
the  equation 

/(*)  =  0(0  +  c. 

If  x,  t  and  x',  t',  denote  two  pairs  of  values  that  are  placed 
opposite  to  one  another,  we  shall  have  simultaneously 

/(*)  =  g(t)  +  c, 

/(*')  =  g(t'}  +  c, 
or  by  eliminating  c 

/(*)- 0(0  =/(*') -<7(0- 

The  ordinary  slide  rule  carries  two  identical  scales  y  =  log  x  and 
y  =  log  t  that  are  able  to  slide  along  one  another,  x  and  t  running 
through  the  values  1  to  100.  We  therefore  have 

log  x  —  log  t  =  log  x'  —  log  t', 
or 

x__  ^ 

t  ~~  t'  '  . 


FIG.  33. 


That  is  to  say,  in  any  position  of  the  x-  and  <-scale  any  two  values 
x  and  t  opposite  each  other  have  the  same  ratio  (Fig.  33).     This 


44  GRAPHICAL   METHODS. 

is  the  principle  on  which  the  use  of  the  slide  rule  is  founded. 
It  enables  us  to  calculate  any  of  the  four  quantities  x,  t,  x',  t' 
if  the  other  three  are  given.  Suppose,  for  example,  x,  t,  x' 
known.  We  set  the  scales  so  that  x  appears  opposite  to  t, 


«,  r— 

—  rt   i  i 

fflSSPi 

20     30  10    '  60  80  100 

T  >      }          if 

?        f      1 

5       6     j     8    flp 

Y7"*"     1— 

JL5          2                3 

i         5      6     f    iUl 

•^ 

Tr 

FIG.  34. 

then  i'  is  read  off  opposite  to  x'.  On  the  other  edges  the  slide 
rule  carries  two  similar  scales  one  double  the  size  of  the  other 
(Fig.  34).  We  may  write 

y  =  2  log  X    and    y  =  2  log  T. 

By  means  of  a  little  frame  carrying  a  crossline  and  sliding  over 
the  instrument,  we  can  bring  the  scales  x  and  T  or  t  and  X  op- 
posite each  other.  If,  for  example,  for  any  position  of  the 
instrument  x,  T  and  x',  T'  are  two  pairs  of  values  opposite  each 
other,  then 

log  x  -  2  log  T  =  log  x'  -  2  log  7", 
or 

x        x 


If  any  three  of  the  four  quantities  x,  T,  x',  T'  are  known  the 
fourth  may  be  read  off.     Thus  we  find  the  value 

xT'2 


by  setting  T  opposite  to  x  and  reading  off  the  value  opposite  to 
T'.    Or  we  can  find  the  value  of 


f-r 

by  setting  a;  opposite  to  T  and  reading  off  the  value  opposite  x'. 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS.        45 

Let  us  reverse  the  part  that  carries  the  scales  t,  T  so  that  x 
slides  along  T  and  X  along  t,  but  in  the  opposite  order  (Fig.  35). 


FIG.  35. 
The  scales  t,  T  may  then  be  expressed  by 

y  =  I  —  log  t    and    y  =  I  —  2  log  T, 

I  being  the  entire  length  of  the  scales. 

By  setting  the  instrument  to  any  position  and  considering  the 
scales  x  and  t  or  X  and  T  by  means  of  the  cross  line  we  have 
log  x  +  log  t  =  log  x'  +  log  t'  and  log  X  +  log  T  =  log  X'  +  log  T 
or 

xt  =  x'tf    and    XT  =  XT, 

so  that  any  two  values  opposite  to  one  another  have  the  same 
product. 

For  x  and  T  we  have 


xT2  =  x'T'2. 
Let  us  apply  this  to  find  the  root  of  an  equation  of  the  form 

w3  +  au  =  6. 
Divide  by  u  so  that 


and  set  T  =  1  opposite  to  X  =  6.  Then  taking  T  =  u  we  find 
on  the  same  cross  line  t  =  v?  and  X  =  b/u,  so  that  we  read  the 
two  values  w2  and  bfu  directly  opposite  to  each  other  on  the 
scales  t  and  X.  If  b/u  is  positive,  it  decreases  while  u2  increases. 


46  GRAPHICAL  METHODS. 

Running  our  eye  along  we  have  to  find  the  place  where  the  differ- 
ence b/u  —  u2  is  equal  to  a.  Having  found  it  the  T-scale  gives 
us  the  root  of  the  equation.  For  example  take 

us  -  5u  =  3, 
or 


We  set  T  =  1  opposite  X  =  3  and  run  our  eye  along  the  scales 
X  and  t  (Fig.  36),  to  find  the  place  where  i  -  5  =  X.    We  find 


2  —  TTT 

/  ct  or    oc 

8 
OT6»: 

9  e  »    s     ? 

'   '    T 

p^- 

FIG.  36. 

it  approximately  at  t  =  6.2,  and  on  the  T-scale  we  read  off 
T  =  2.50  as  the  approximate  value  of  the  root.  This  is  the 
only  positive  root.  But  for  a  negative  root  3/w  is  negative, 
and  therefore  the  positive  value  of  3/w  plus  w2  would  have  to  be 
equal  to  5.  We  run  our  eye  along  and  find  t  =  3.37  opposite  to 
X  =  1.63,  approximately  corresponding  to  T  =  1.84.  There- 
fore —  1.84  is  another  root.  As  the  coefficient  of  u2  in  the  first 
form  of  the  equations  vanishes  it  follows  that  the  sum  of  the 
three  roots  must  be  equal  to  zero.  This  demands  a  second 
negative  root  approximately  equal  to  —  0.66.  To  make  sure 
that  it  is  so,  we  set  the  instrument  back  and  take  the  other  end 
of  the  T-scale  as  representing  the  value  T  =  1  and  give  it  the 
position  this  end  had  before.  Running  our  eye  along  the 
scales  X  and  t,  we  find  t  =  0.43  opposite  to  X  =  4.57,  giving 
X  +  t  =  5.00.  On  the  f-scale  we  find  0.655,  so  that  the  third 
root  is  found  equal  to  —  0.655. 

When  b  is  negative  there  is  always  one  and  only  one  negative 
root.  For  u  running  through  the  values  u  =  0  to  —  oo,  u2—b/u 
will  run  from  —  oo  to  +  oo  without  turning.  When  b  is  positive 
there  is  always  one  and  only  one  positive  root;  for  then  u?  —  b/u 


GRAPHICAL  REPRESENTATION   OF   FUNCTIONS.  47 

runs  from  —  oo  to  +  oo  for  u  =  0  to  +  oo.  In  the  first  case 
there  may  be  two  positive  roots  or  none;  in  the  second  case  there 
may  be  two  negative  roots  or  none.  For  positive  values  of  a 
one  root  only  exists  in  either  case.  This  is  easily  seen  in  the  first 
form  of  the  equation 

uz  +  au  =  b, 

because  from  a  positive  value  of  a  it  follows  that  u3  +  au  will 
for  M  =  —  oo  to  +  oo,  run  from  —  oo  to  +  °°  without  turning 
and  will  therefore  pass  any  given  value  once  only. 

In  order  to  decide  whether  in  the  case  of  a  negative  value  of  a 
there  are  three  roots  or  only  one  let  us  write 

U?  —  -  =  —  a. 
u 

For  negative  values  of  b  we  have  to  investigate  whether  there 
are  positive  roots.  For  positive  values  of  u  the  function  u?—bfu 
has  a  minimum,  when  the  differential  coefficient  vanishes,  i.  e.,  for 


Having  set  our  slide  rule  so  that  t  gives  us  w2  and  X  gives  us 
—  b/u,  we  find  the  value  u  where  the  minimum  takes  place  by 
running  our  eye  along  and  looking  for  the  values  X,  t  opposite 
each  other  for  which  X  is  twice  the  value  of  t 

2t  =  X. 

Then  t  +  X  is  the  minimum  of  uz  —  b/u,  so  that  there  will  be 
two  or  no  positive  roots  according  to  t  +  X  being  smaller  or 
larger  than  —  a.  For  positive  values  of  b,  we  have  to  find  out 
whether  there  are  negative  roots.  The  criterion  is  the  same. 
After  having  set  T  =  1  opposite  to  6  and  having  found  the 


48  GRAPHICAL  METHODS. 

positive  root,  we  find  the  place  where 
2t=  X. 

Then  t  +  X  is  the  minimum  of  all  values  that  uz  —  bfu  assumes 
for  negative  values  of  u.     If  the  minimum  is  smaller  than  —  a 
there  are  two  negative  roots;  if  it  is  larger  there  are  none.     If  it 
is  equal  to  —  a  the  two  negative  roots  coincide. 
For  the  equation 


for  instance,  we  find  t  =  1.31  opposite  to  X  =  2.62  (Fig.  36), 
so  that  2t  =  2.62  =  X.  Now  t  +  X  =  3.93  is  smaller  than  5, 
therefore  u2  —  3/w  will  assume  the  value  5  for  two  negative 
values  of  u  on  either  side  of  the  value  u  =  —  T  =  —  1.143 
for  which  the  minimum  of  u2  —  3/w  takes  place. 

On  the  same  principle  as  the  slide  rule  many  other  instruments 
may  be  constructed  for  various  calculations.  In  all  these  cases 
we  have  for  any  position  of  the  instrument 

/(*)-  0(0  =/(*')  -0(O, 

where  x,  t  are  any  readings  of  the  two  scales  opposite  each  other 
and  x't'  the  readings  at  any  other  place.  f(x)  and  g(f)  may  be 
any  functions  of  x  and  t.  It  will  only  be  desirable  that  they 
be  limited  to  intervals  of  x  and  t,  which  contain  no  turning 
points.  Else  the  same  point  of  the  scale  corresponds  to  more 
than  one  value  of  x  or  t  and  that  will  prevent  a  rapid  reading 
of  the  instrument. 

Let  us  design  an  instrument  for  the  calculation  of  the  increase 
of  capital  at  compound  interest  at  a  percentage  from  2  per  cent. 
upward.  If  x  is  the  number  of  per  cent,  and  t  the  number  of 
years,  the  increase  of  capital  at  compound  interest  is  in  the  pro- 
portion 


EEPRESENTATION  OF  FUNCTIONS.        49 

We  can  evidently  build  an  instrument  for  which 


For  taking  first  the  logarithm  and  then  the  logarithm  of  the 
logarithm,  we  obtain 


We  have  only  to  make  the  ar-scale 

y  =  +  loglog(l  +  j^)-  log  log  (l  + 

and  the  2-scale 

y  =  log  n  —  log  t. 

For  x  =  2  we  have  y  =  0  and  therefore  in  the  normal  position 
of  the  instrument  t  —  n.  On  the  other  end  we  have  t  =  1  and 
therefore  y  =  log  n.  Now  let  us  take  n  =  100,  so  that  y  =  2 
for  t  =  1.  Say  the  length  of  the  instrument  is  to  be  about  24 
cm.,  then  the  unit  of  length  for  the  jr-scale  would  have  to  be  12 
cm.  In  the  normal  position  of  the  instrument  the  readings  x,  t 
opposite  to  each  other  satisfy  the  equation 


Opposite  t  =  1,  we  read  the  value  x\  =  624  and  this  gives  us 


A  capital  will  increase  in  100  years  at  two  per  cent,  compound 
interest  in  the  proportion  7.24  :  1.  Or  we  may  also  say  the 
number  x\  =  624  read  off  opposite  t  =  1  is  the  amount  which  is 
added  to  a  capital  equal  to  100  by  double  interest  of  2  per  cent. 
in  100  years.  The  same  position  of  the  instrument  gives  us  the 
number  of  years  that  are  wanted  for  the  same  increase  of  capital 
5 


50  GRAPHICAL  METHODS. 

at  a  higher  percentage.     For  all  the  values  x,  t  opposite  to  each 
other  satisfy  the  equation 


For  any  other  given  percentage  x  and  any  other  given  number 
of  years  t  the  increase  of  capital  is  found  by  setting  x  opposite 
to  t  and  reading  the  z-scale  opposite  to  t  =  1.  The  only  restric- 
tion is  that  the  ratio  is  not  greater  than  7.24,  else  t  =  1  will 
lie  beyond  the  end  of  the  or-scale. 

For  a  given  increase  of  capital  the  instrument  will  enable  us 
either  to  find  the  number  of  years  if  the  percentage  is  given,  or 
the  percentage  if  the  number  of  years  is  given,  subject  only  to 
the  restriction  mentioned. 

We  can  build  our  instrument  so  as  to  include  greater  increases 
of  capital  by  choosing  a  larger  value  of  n.  n  =  1000,  for  in- 
stance, will  make  y  =  3  for  t  =  1.  If  the  instrument  is  not  to 
be  increased  in  size  the  scales  would  have  to  be  reduced  in  the 
proportion  2  :  3. 

Let  us  consider  another  instance 

1  1       1 


In  the  normal  position  of  the  instrument  the  scale  division 
marked  x  =  «D  corresponds  to  y  =  0  and  is  opposite  to  t  =  n. 
If  we  have  t  =  oo  on  the  other  end,  the  length  of  the  instrument 
will  correspond  to  y  =  1/n.  Let  us  choose  n  =  0.1,  so  that  the 
length  of  the  instrument  is  y  =  10.  That  is  to  say,  the  unit  of 
length  of  the  y-scale  is  one  tenth  of  the  length  of  the  instrument. 
For  any  position  of  the  instrument  we  have 

-+---+- 

x+  t~  x'^f 

If  the  scale  division  marked  x  =  oo  is  opposite  to  t  =  c  we  can 
write  x'  =  oo,t'  =  c  and  have 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 


51 


1+-  =  -. 

X^  t    C 

The  instrument  will  therefore  enable  us  to  read  off  any  one  of 
the  three  quantities  x,  t,  c,  if  the  other  two  are  given,  the  only 
restriction  being  that  all  three  lie  within  the  limits  0.1  to  oo. 
The  instrument  may  be  used  to  determine  the  combined  resistance 
of  two  parallel  electrical  re- 
sistances, for  the  resistances 
satisfy  the  equation 


- 
R,' 


FIG.  37. 


Similarly  it  may  be  used 
to  calculate  the  distances  of  an  object  and  its  image  from  the 
principal  planes  of  any  given  system  of  lenses.  For  if  /  is  the 
focal  length  and  x  and  t  the  distances  of  the  object  and  its  im- 
age from  the  corresponding  principal  planes  (Fig.  37),  the  equa- 
tion is 

-+--1- 

x+t      f 

On  the  back  side  of  the  movable  part  of  an  ordinary  slide  rule 
there  generally  is  a  scale 

y  =  2  +  log  sin  t. 

When  this  part  is  turned  round  and  the  scale  is  brought  into 
contact  with  the  scale 

y  =  log  x, 

we  obtain  for  any  position  of  the  instrument 

log  x  —  log  sin  t  =  log  x'  —  log  sin  t', 


or 


x  x' 

sin  t       sin  t' ' 


for  any  two  pairs  of  values  x,  t  that  are  opposite  each  other. 


52  GRAPHICAL  METHODS. 

Given  two  sides  of  a  triangle  and  the  angle  opposite  the  larger 
of  the  two  the  instrument  gives  at  once  the  angle  opposite  the 
other  side.  Similarly  when  two  angles  and  one  side  are  given, 
it  gives  the  length  of  the  other  side. 

If  x'  =  a  is  the  value  opposite  to  t'  =  90°,  we  have 

x  =  a  sin  t. 

Thus  we  can  read  the  position  of  any  harmonic  motion  for  any 
value  of  the  phase. 

An  instrument  carrying  the  scales 

y  —  log  sin  x    and    y  =  log  sin  t 
enables  us  to  find  any  one  of  four  angles  x,  t,  x',  t'  for  which 

sin  x  _  sin  x' 
sin  t       sin  t' 

if  the  other  three  are  given.  Thus,  knowing  the  declination, 
hour  angle  and  height  of  a  celestial  body,  we  can  read  the  azimuth 
on  the  instrument.  We  have  only  to  take  x  =  90°  —  height, 
t  =  hour  angle,  x'  =  90°  —  declination,  then  t'  =  azimuth  or 
180°  —  azimuth. 

It  is  not  necessary  to  carry  out  the  subtraction  90°  —  height  and 
90°  —  declination.  The  difference  may  be  counted  on  the  scale 
by  imagining  0°  written  in  the  place  of  90°,  10°  in  the  place  of 
80°  and  so  on  and  counting  the  partitions  of  the  scale  backwards 
instead  of  forward. 

§  8.  Rectangular  Coordinates  with  Intervals  of  Varying  Size. — 
The  two  methods  of  representing  the  relation  between  two 
variables  either  by  a  curve  connecting  the  coordinates  or  by 
scales  facing  each  other  lead  to  a  combination  of  both. 

Suppose  the  rectangular  coordinates  x  and  y  are  functions  of 
u  and  v, 

x  =  <f>(u)     and    y  =  t(x). 

The  function  x  =  <p(u)  is  represented  by  a  uniform  scale  for  x 
on  the  axis  of  abscissae  facing  a  non-uniform  scale  for  u.  The 


GRAPHICAL   REPRESENTATION   OF   FUNCTIONS. 


53 


function  y  =  ^(c)  is  represented  by  a  uniform  scale  for  y  on 
the  axis  of  ordinates  facing  a  non-uniform  scale  for  v.  Through 
the  scale-divisions  u  let  us  draw  vertical  lines,  and  through  the 
scale-divisions  v  let  us  draw  horizontal  lines.  These  two  systems 
of  parallel  lines  form  a  network  of  rectangular  meshes  of  various 
sizes  (Fig.  38),  and  any  equation  between  u  and  v  may  be  repre- 
sented by  a  curve  in  this  plane. 

The  usefulness  of  this  method  will  be  seen  by  some  examples. 
It  enables  us  by  a  clever  choice  of  the  functions  <p(u)  and  $(v) 


> 


FIG.  38. 


123       i      5 
FIG.  39. 


to  simplify  the  form  of  the  curve.  It  is  easily  seen,  for  instance, 
that  a  curve  representing  an  equation /(w,  v)  =  0  may  always  be 
replaced  by  a  straight  line,  if  we  choose  the  w-scale  properly. 
For  when  the  points  u  =  1,  2,  3,  4,  •  •  •  of  the  curve  are  not  on 
a  straight  line,  let  them  be  moved  to  a  straight  line  without 
altering  their  ordinates  (Fig.  39).  This  will  change  the  w-scale 
but  it  will  not  alter  the  equation  f(u,  «)  =  0  now  represented  by 
the  straight  line. 

Suppose  we  want  to  represent  the  relation 


where  a  and  6  are  given  numbers.  If  u  and  v  were  ordinary 
rectangular  coordinates  the  curve  would  be  an  ellipse.  But  if  we 
make 

x  =  u2    and     v  =  v* 


54  GRAPHICAL  METHODS. 

the  equation  of  the  line  in  rectangular  coordinates  becomes 


and  the  curve  will  therefore  be  a  straight  line  running  from  a 
point  on  the  positive  axis  of  x  to  a  point  on  the  positive  axis  of 
y.  The  point  on  the  axis  of  x  corresponds  to  the  value  u  =  =*=  a 

on  the  w-scale,  and  the 
point  on  the  axis  of  y  cor- 
responds to  the  value  v  = 
=t  b  on  the  r-scale  (Fig.  40). 
Any  point  on  the  straight 
line  corresponds  to  four 
combinations  -j-w,  +0;— u, 
+  v,  u,  —  v;  —  u,  —  v,  be- 
cause x  has  the  same  values 
for  opposite  values  of  u 
We  can  read  v  as  a  function  of 


±2.5 
±2 

±1.5 
±1 


-J.-.L.O  _*   irf.a     io       rj 


FIG.  40. 


and  y  for  opposite  values  of 
u  or  u  as  a  function  of  v. 
If  a  second  equation 


is  given,  we  find  the  common  solutions  of  the  two  equations  by 
the  intersection  of  the  corresponding  straight  lines.  Fig.  40 
shows  the  solutions  of  the  two  equations 


1 


and 


*       2V 

42  -I-  52   ~ 


approximately  equal  to  u  =  =*=  1.2  and  0  =  ±  2.4. 
Another  function  much  used  in  mathematical  physics 


_ 

v  =  ae  m* 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 


55 


may  also  be  represented  by  a  straight  line  by  means  of  the  same 
device. 
By  making 

y  —  log  v,     x  =  u2, 
we  obtain 


where  log  v  and  log  a  are  the  natural  logarithms  of  v  and  a. 
The  i/-scale  is  laid  off  on  the  axis  of  x  and  the  u-scale  on  the  axis 
of  y  and  we  have  to  join  the  points  u  =  0,  v  =  a  and  u  =  m, 
v  =  a/e.  The  point  v  =  a/e  is  found  by  laying  off  the  distance 
v  =  1  to  v  =  e  from  v  =  a  downward  (Fig.  41).  We  are  not 
obliged  to  take  the  same  units  of  length  for  x  and  y. 


FIG.  41. 

Suppose  we  had  to  find  the  constants  a  and  m  from  two  equa- 
tions 


and 


_o 

=  ae  ^' 


Our  diagram  would  furnish  two  points  corresponding  to  u\,  vi 
and  u%,  TZ.  The  straight  line  joining  these  two  points  intersects 
the  axis  of  ordinates  at  v  =  a  and  intersects  the  parallel  through 
x  =•  a/e  to  the  axis  of  abscissae  at  u  =  m. 


56 


GRAPHICAL  METHODS. 


In  applied  mathematics  the  problem  would  as  a  rule  present 
itself  in  such  a  form  that  more  than  two  pairs  of  values  u,  v 
would  be  given  but  all  of  them  affected  with  errors  of  observation. 
The  way  to  proceed  would  then  be  to  plot  the  corresponding 
points  and  to  draw  a  straight  line  through  the  points  as  best  we 
can.  A  black  thread  stretched  over  the  drawing  may  be  used  to 
advantage  to  find  a  straight  line  passing  as  close  to  the  points 
as  possible  (Fig.  42). 

In  several  other  cases  the  variables  u  and  v  are  connected  with 
the  rectangular  coordinates  x  and  y  by  the  functions 

x  =  log  u    and    y  =  log  v. 


4' 


Fio.  43. 


"Logarithmic  paper"  prepared  with  parallel  lines  for  equidistant 
values  of  u  and  lines  perpendicular  to  these  for  equidistant  values 
of  v  is  manufactured  commercially  (Fig.  43). 

By  this  device  diagrams  representing  the  relation 

urv*  =  c, 

where  r,  s,  c  are  constants  are  given  by  straight  lines.  For  by 
taking  the  logarithm  we  obtain 

rx  +  sy  =  log  c. 

The  straight  line  connects  the  point  u  =  e1/r  on  the  w-scale  with 
the  point  v  =  cl/t  on  the  fl-scale. 

Logarithmic  paper  is  further  used  to  advantage  in  all  those 


GRAPHICAL   REPRESENTATION    OF   FUNCTIONS.  57 

cases  where  a  variety  of  relations  between  the  variables  u  and  v 
are  considered  that  differ  only  in  u  and  v  being  changed  in  some 
constant  proportion.  If  u  and  v  were  plotted  as  rectangular 
coordinates  the  curves  representing  the  different  relations  be- 
tween u  and  v  might  all  be  generated  from  one  of  them  by  altering 
the  scale  of  the  abscissae  and  independently  the  scale  of  the  ordi- 
nates,  so  that  the  appearance  of  all  these  curves  would  be  very 
different.  Let  us  write 

M  «)  =  o, 

as  the  equation  of  one  of  the  curves.  The  equations  of  all  the 
rest  may  then  be  written 


where  a,  b  are  any  positive  constants.  The  points  u,  v  of  the 
first  curve  lead  to  the  points  on  one  of  the  other  curves  by  taking 
u  a  times  as  great  and  v  b  times  as  great.  For  if  we  write  u'  =  au 
and  v'  =  bu  the  equation  f(u,  v)  =  0  leads  to  the  equation 
between  u'  and  v': 


Using  logarithmic  paper  the  diagram  of  all  these  curves  be- 
comes very  much  simpler.  The  equation /(w,  v)  =  0  is  equivalent 
to  a  certain  equation  <p(x,  y)  =  0,  where  x  =  log  u,  y  =  log  i>. 
Now  let  x',  y'  be  the  rectangular  coordinates  corresponding  to 
u',  v'  so  that 

x'  —  log  u'  =  log  u  +  log  a  =  x  +  log  a, 
y'  =  log  v'  =  log  v  +  log  6  =  y  +  log  b. 

The  point  x',  y'  is  reached  from  the  point  x,  y  by  advancing 
through  a  fixed  distance  log  a  in  the  direction  of  the  axis  of  x 
and  a  fixed  distance  log  b  in  the  direction  of  the  axis  of  y.  The 
whole  curve 

f(u,  v)  =  0 


58  GKAPHICAL  METHODS. 

drawn  on  logarithmic  paper  is  therefore  identical  with  all  the 


It  can  be  made  to  coincide  with  any  one  of  the  curves  by 
moving  it  along  the  directions  of  x  and  y. 

§  9.  Functions  of  Two  Independent  Variables. — When  a  func- 
tion of  one  variable  y  =  f(x)  is  represented  by  a  curve,  the  values 
of  x  are  laid  off  on  the  axis  of  x  and  the  values  of  y  are  represented 
by  lines  perpendicular  to  the  axis  of  x.  In  a  similar  way  a 
function  of  two  independent  variables 

*  =  /(*,  y} 

may  be  represented  by  plotting  x  and  y  as  rectangular  coordinates 
and  erecting  lines  perpendicular  to  the  xy  plane,  in  all  the 
points  x,  y,  where  f(x,  y)  is  defined  and  making  the  lengths  of 
the  perpendiculars  proportional  to  2.  In  this  way  the  function 
corresponds  to  a  surface  in  space*  Now  there  are  practical 
difficulties  in  working  with  surfaces  in  space  and  therefore  it 
appears  desirable  to  use  other  methods,  that  enable  us  to  represent 
functions  of  two  independent  variables  on  a  plane.  This  may 
be  done  in  the  following  way. 

Taking  x,  y  as  rectangular  coordinates  all  the  points  for  which 
f(x,  y)  has  the  same  value  form  a  curve  in  the  xy  plane.  Let 
us  suppose  a  number  of  these  curves  drawn  and  marked  with  the 
value  of  f(x,  y}.  If  the  different  values  of  f(x,  y)  are  chosen 
sufficiently  close,  so  that  the  curves  lie  sufficiently  close  in  the 
part  of  the  xy  plane  that  our  drawing  comprises,  we  are  not  only 
able  to  state  the  value  of  f(x,  y}  at  any  point  on  one  of  the  drawn 
curves,  but  we  are  also  able  to  interpolate  with  a  certain  degree 
of  accuracy  the  value  of  f(x,  y}  at  a  point  between  two  of  the 
curves.  As  a  rule  it  will  be  convenient  to  choose  equidistant 
values  of  f(x,  y)  to  facilitate  the  interpolation  of  the  values 
between.  The  curves  may  be  regarded  as  the  perpendicular 
projection  of  certain  curves  on  the  surface  in  space,  the  inter- 


GRAPHICAL  REPRESENTATION   OF  FUNCTIONS.  59 

sections  of  the  surface  by  equidistant  planes  parallel  to  the 
xy  plane. 

The  method  is  the  generalization  of  the  scale-representation 
of  a  function  of  one  variable.  For  a  relation  between  t  and  x 
represented  by  a  curve  with  t  as  ordinate  and  x  as  abscissa,  is 
transformed  into  a  scale  representation  by  perpendicularly 
projecting  certain  points  of  the  curve  onto  the  axis  of  x,  the 
intersections  of  the  curve  by  equidistant  lines  parallel  to  the  axis 
of  x  and  marking  them  with  the  value  of  t.  A  scale  division  in 
the  case  of  a  function  of  one  variable  corresponds  to  a  curve  in 
the  case  of  a  function  of  two  independent  variables. 

This  method  of  representing  a  function  of  two  independent 
variables  by  a  plane  drawing  or  we  might  also  say  of  representing 
a  surface  in  space  by  a  plane  drawing,  is  used  by  naval  architects 
to  render  the  form  of  a  ship  and  by  surveyors  to  render  the  form 
of  the  earth's  surface  and  by  engineers  generally.  Let  us  apply 
the  method  to  a  problem  of  pure  mathematics. 

The  equation  q-   -p?_ -i.3 

£  +  pz  +  q  =  0 

defines  z  as  a  function  of  p  and  q.  Let  us  represent  this  function 
by  taking  p  and  q  as  rectangular  coordinates  and  drawing  the 
lines  for  equidistant  values  of  z. 

For  any  constant  value  of  z  we  have  a  linear  equation  between 
the  variables  p  and  q,  and  therefore  it  is  represented  by  a  straight 
line.  This  line  intersects  the  parallels  p  =  1  and  p  =  —  1  at 
the  points  q  =  —  z3  —  z  and  q  =  —  z3  +  z.  Let  us  calculate 
these  values  for  z  =  0;  ±0.1;  ±  0.2  •  •  •  ±  1.3  and  in  this  way 
draw  the  lines  corresponding  to  these  values  of  z  as  far  as  they 
lie  in  a  square  comprising  the  values  p  =  —  1  to  +  1  and 
q  =  —  1  to  +  1.  Fig.  44  shows  the  result.  On  this  diagram 
we  can  at  once  rsad  the  roots  of  any  equation  of  the  third  degree 
of  the  form 

z3  +  pz  +  q  =  0, 

where  p  and  q  lie  within  the  limits  —  1  to  +  1.     For  p  =  0.4  and 


60 


GRAPHICAL  METHODS. 


q  —  —  0.2,  for  instance,  we  read  z  =  0.37,  interpolating  the  value 
of  z  according  to  the  position  of  the  point  between  the  lines 
z  =  0.3  and  z  =  0.4.  We  also  see  that  there  is  only  one  real 
root,  for  there  is  only  one  straight  line  passing  through  the  point. 


-1.3 


0.3 


FIG.  44. 

On  the  left  side  of  the  square  there  is  a  triangular-shaped  region 
where  the  straight  lines  cross  each  other.  To  each  point  within 
this  region  corresponds  an  equation  with  three  real  roots.  For 
example,  at  the  point  p=  —  0.8  and  q  =  +  0.2  we  read  z  = 
—  1.00;  +  0.28;  +  0.72.  On  the  border  of  this  region  two  roots 
coincide. 

For  values  of  p  and  q  beyond  the  limits  —  1  to  +  1  the  diagram 
may  also  be  used.  We  only  have  to  introduce  z'  =  z/ra  instead 
of  z  and  to  choose  m  sufficiently  large. 

Instead  of 

z3  +  pz  +  q  =  0 


GRAPHICAL   REPRESENTATION    OF   FUNCTIONS.  61 

we  obtain 

mV3  +  pmz'  +  q  =  0, 
or  dividing  by  m3, 


where 

*'-&  '-£• 

By  choosing  a  sufficiently  large  value  of  m,  p'  and  g'  can  be 
made  to  lie  within  the  limits  —  1  to  -+-  1  so  that  the  roots  z' 
may  be  read  on  the  diagram.  Multiplying  them  by  m  we 
obtain  the  roots  z  of  the  given  equation. 

A  function  of  two  independent  variables  need  not  be  expressed 
in  an  explicit  form,  but  may  be  given  in  the  form  of  an  equa- 
tion between  three  variables 

g(u,  0,  w}  =  0, 

and  we  may  consider  any  two  of  them  as  independent  and  the 
third  as  a  function  of  the  two.  The  graphical  representation 
may  sometimes  be  greatly  facilitated  by  modifying  the  method 
described  before.  The  curves  for  constant  values  of  one  of  the 
three  variables,  say  w,  are  not  plotted  by  taking  u  and  v  as 
rectangular  coordinates,  but  they  are  plotted  after  introducing 
new  variables  x  and  y,  x  a  function  of  u  and  y  a  function  of  v  and 
making  x  and  y  the  rectangular  coordinates. 

In  some  cases,  for  instance,  we  can  succeed  by  a  right  choice 
of  the  functions  x  =  <p(u)  and  y  =  $(v)  in  getting  straight  lines 
for  the  curves  w  =  const.  This  will  evidently  be  the  case, 
when  the  equation  g(u,  v,  w)  =  0  can  be  brought  into  the  form 

a(w~)<p(u)  +  b(w}\l/(/e)  +  c(u')  =  0, 

a,  b,  c  being  any  functions  of  w,  <f>  any  function  of  u  and  \f/  any 
function  of  v. 
For  introducing 


62  GRAPHICAL  METHODS. 

*  =  <t>(u),    y  =  t(v) 
the  equation  will  become 

ax  +  by  +  c  =  0, 

where  a,  b,  c  are  constants  for  any  constant  value  of  w. 

As  an  example  let  us  consider  the  relation  between  the  true 
solar  time,  the  height  of  the  sun  over  the  horizon,  and  the  declina- 
tion of  the  sun  for  a  place  of  given  latitude.  Instead  of  the 
declination  of  the  sun  we  might  also  substitute  the  time  of  the 
year,  as  the  time  of  the  year  is  determined  by  the  declination  of 
the  sun.  Our  object  then  is  to  make  a  diagram  for  a  place  of 
given  latitude  from  which  for  any  time  of  the  year  and  any 
height  of  the  sun  the  true  solar  time  may  be  read. 

In  the  spherical  triangle  formed  by 
the  zenith  Z,  the  north  pole  P  (if  we  sup- 
pose the  place  to  be  on  the  northern 
hemisphere)  and  the  sun  S  (Fig.  45),  the 
sides  are  the  complements  of  the  decli- 
nation 6,  the  height  h,  and  the  latitude 
<p.  The  angle  t  at  the  pole  is  the  hour 
angle  of  the  sun,  which  expressed  in 
time  gives  true  solar  time. 

The  equation  between  these  four  quantities  may  be  written  in 
the  form 

sin  h  =  sin  <p  sin  5  +  cos  <f>  cos  5  cos  t. 

The  latitude  <p  is  to  be  kept  constant,  so  that  t,  h,  5  are  the  only 
variables. 
Now  let  us  write 

x  =  cos  t,    y  =  sin  h, 
so  that  the  equation  takes  the  form 

y  =  sin  <p  sin  8  +  %  cos  <p  cos  5. 
When  x  and  y  are  plotted  as  rectangular  coordinates,  we  obtain 


GKAPHICAL   REPRESENTATION   OF  FUNCTIONS. 


63 


a  straight  line  for  any  value  of  5.  Let  us  draw  horizontal  lines 
for  equidistant  values  of  h  =  0  to  90°  and  vertical  lines  for  equi- 
distant values  of  t  =  —  180°  to  +  180°  or  expressed  in  time 
from  midnight  to  midnight  (Fig.  46).  In  order  to  draw  the 


latitude  =  O. 

FIG.  46. 

straight  lines  8  =  const.,  let  us  calculate  where  they  intersect 
the  vertical  lines  corresponding  to  x  =  —  1  and  x  =  +  1  or 
expressed  in  time  corresponding  to  midnight  and  to  noon.  For 
x  =  —  1  we  have  y  =  —  cos  (<p  +  6),  and  for  x  =  +  1  we  have 
y  =  cos  (<p  —  5).  Let  us  draw  a  scale  on  the  vertical  x  =  —  1 
showing  the  points  y  =  —  cos  (p  +  5)  for  equidistant  values  of 
(<p  +  6)  and  a  scale  on  the  vertical  x  =  +  1,  showing  the  points 
y  —  cos  (<f>  —  5)  for  equidistant  values  of  <p  —  8.  The  scale  is 
the  same  as  the  scale  for  h,  with  the  sole  difference  that  the  values 
of  <p  —  8  are  the  complements  of  h  and  the  values  of  <p  +  8  the 
complements  of  —  h.  For  a  latitude  of  41°,  for  instance,  we 

have 

For  S  <p  +  s  <?  -  d 

June  21 23.5°  64.5°  17.5° 

September  23  and  March  21 0  41°  41° 


December  21 -23.5° 


17.5° 


64.5e 


64 


GRAPHICAL   METHODS. 


The  values  of  <p  +  5  and  <p  —  6  furnish  the  intersections  with 
the  verticals  x  =  —  1  and  x  =  +  1,  so  that  the  straight  lines 
can  be  drawn  corresponding  to  these  days  of  the  year.  The  two 
outward  lines  are  parallel  but  the  middle  line  is  steeper.  Their 
intersections  with  the  horizontal  line  h  —  0  show  the  time  of 
sunrise  and  sunset.1  Strictly  speaking  the  straight  lines  do 
not  correspond  to  certain  days.  The  straight  line  determined 
by  any  value  of  8  changes  its  position  continually  as  5  changes 
continually.  But  the  changes  of  5  during  one  day  are  scarcely 
appreciable  unless  the  drawing  is  on  a  larger  scale. 
If  in  the  equation 

ax  +  by  -f  c  =  0 

a  and  b  are  independent  of  w,  only  c  being  a  function  of  w,  all 
the  straight  lines  w  =  const,  are  parallel.     In  this  case  we  are 

not  obliged  to  draw  the 
straight  lines  w  =  const. 
It  will  suffice  to  draw  a 
line  perpendicular  to  the 
lines  w  =  const,  and  a 
scale  on  it  that  marks  the 
points  corresponding  to 
equidistant  values  of  w. 
On  the  drawing  we  place  a 
sheet  of  transparent  paper 
or  celluloid,on  which  three 
straight  lines  are  drawn  is- 
suing from  one  point  in  the  direction  perpendicular  to  the  w-scale, 
7>-scale  and  w-scale  (Fig.  47).  If  we  move  the  transparent  material 
without  turning  it  and  make  the  first  two  lines  intersect  the  u-and-v 
scale  at  given  points,  the  w-scale  will  be  intersected  at  the  point 
corresponding  to  the  value  of  w.  This  method  has  the  advantage 

1  That  is  to  say,  the  moment  when  the  center  of  the  sun  would  be  seen  on 
the  horizon,  if  there  were  no  atmospherical  refraction.  To  take  account  of 
the  refraction,  the  line  h  =  —  0.6°  would  have  to  be  considered  instead  of 
h  =  0. 


FIG.  47. 


GRAPHICAL   REPRESENTATION   OF  FUNCTIONS.  65 

that  we  can  use  the  same  paper  for  a  great  many  relations  of 
three  variables,  as  we  can  place  a  great  many  scales  side  by  side. 
Or,  in  the  case  of  one  relation  only,  we  may  divide  the  region  of 
the  values  u,  v,  w  into  a  number  of  smaller  regions  and  draw  three 
scales  for  each  of  them,  placing  all  the  w-scales  or  ^-scales  or 
w/'-scales  side  by  side.  The  drawing  will  then  have  the  same 
accuracy  as  a  drawing  of  very  much  larger  size  in  which  there 
is  only  one  scale  for  each  of  the  three  variables. 

§  10.  Depiction  of  One  Plane  on  Another  Plane. — Let  us  now 
consider  two  quantities  x  and  y  each  as  a  function  of  two  other 
quantities  u  and  v 

x  =  <p(u,  r), 

y  =  \l/(u,  T). 

In  order  to  give  a  geometrical  meaning  to  this  relation  between 
two  pairs  of  quantities  let  us  consider  x  and  y  as  rectangular 
coordinates  of  a  point  in  a  plane  and  u,  v  as  rectangular  coordi- 
nates of  a  point  in  another  plane.  We  then  have  a  corre- 
spondence between  the  two  points.  When  the  functions  (p(u,  rs) 
and  \lt(u,  v)  are  defined  for  the  values  u,  v  of  a  certain  region, 
they  will  furnish  for  every  point  u,  v  of  this  region  a  point  in 
the  xy  plane.  Let  us  call  this  a  depiction  of  the  uv  plane  on 
the  xy  plane.  Similarly  a  function  of  one  variable  x  =  <p(u) 
might  be  said  to  depict  the  u  line  on  the  x  line.  We  may  there- 
fore say  that  the  depiction  of  one  plane  on  another  plane  is,  in 
a  certain  way,  the  generalization  of  the  idea  of  a  function  of  one 
variable.  Let  us  suppose  <p(u,  v)  and  \f/(u,  v)  both  to  have  only 
one  value  for  given  values  of  u  and  v  for  which  they  are  defined. 
Then  there  will  be  only  one  point  in  the  xy  plane  corresponding 
to  a  given  point  in  the  uv  plane.  But  to  a  given  point  in  the 
xy  plane  there  may  very  well  correspond  several  points  in  the 
uv  plane. 

Let  us  try  to  explain  this  by  a  graphical  representation  of  the 
depiction  of  planes  on  each  other.  For  this  purpose  we  draw 
the  curves  x  =  const,  and  y  =  const,  in  the  uv  plane  for  equi- 


66 


GRAPHICAL  METHODS. 


distant  values  of  x  and  y.  In  the  xy  plane  they  correspond  to 
equidistant  lines  parallel  to  the  axis  of  x  and  to  the  axis  of  y. 
The  point  of  intersection  of  two  lines  x  =  a  and  y  =  b  corre- 
sponds to  the  points  of  intersection  of  the  curves 

<f>(u,  v)  =  a    and     \l/(u,  v)  =  b, 

in  the  uv  plane.  If  in  a  certain  region  of  the  uv  plane,  that 
we  consider,  they  intersect  only  once  there  is  only  one  point  in 
the  region  of  the  uv  plane  considered  and  one  point  in  the  xy 
plane  corresponding  to  each  other.  Fig.  48  shows  the  depiction 
of  part  of  the  uv  plane  on  part  of  the  xy  plane.  We  have  a  net 
of  square-shaped  meshes  in  the  xy  plane  and  corresponding  is  a 
net  of  curvilinear  meshes  in  the  uv  plane. 

Let  us  consider  the  curves  x  =  const,  in  the  uv  plane  as  the 
perpendicular  projections  of  curves  of  equal  height  on  a  surface 
extended  over  that  part  of  the  uv  plane.  From  any  point  P 
of  the  surface  corresponding  to  the  values  u,  v  we  proceed  an 


iro.7 


0.2  0.3  0.4  0.5  0.6 


FIG.  48. 


infinitely  small  distance,  u  changing  to  u  -f-  du,  v  to  v  +  dv  and 
x  to  x  +  <&c,  where 

7         dv>  ,     ,   d<p  , 


Let  us  write 


du  =  cos  ads,    dv  =  sin  ads, 


where  ds  signifies  the  length  of  the  infinitely  small  line  from 
u,  v  to  u  +  du,  v  +  dv  in  the  uv  plane  and  a  the  angle  its  direc- 


GRAPHICAL   REPRESENTATION    OF   FUNCTIONS.  67 

tion  forms  with  the  positive  axis  of  x.  Let  PN  be  a  straight  line 
whose  projections  on  the  u  and  v  axis  are  equal  to  d<p/du  and 
d(p/dv  and  let  us  write 


-=rcosX,     ^-= 

r  being  the  positive  length  of  PN  and  X  the  angle  between  its 
direction  and  the  positive  axis  of  x.     Then  we  have 

dx  =  —  du  +  —  dv  =  rds  cos  (a  —  X), 

(jit  uV 

or 

dx 


dx/ds  measures  the  steepness  of  the  ascent.  It  is  positive  when 
the  direction  leads  upward  and  negative  when  it  leads  downward 
and  its  value  is  equal  to  the  tangent  of  the  angle  of  the  ascent. 
From  the  equation 

~r  =  r  cos  (a  —  X) 

we  see  that  the  ascent  is  steepest  for  a  =  X,  where  dx/ds  =  r. 
The  line  PN  in  the  u,  0-plane  shows  the  perpendicular  projection 
of  the  line  of  steepest  ascent  on  the  surface  x  =  <p(u,  v)  and  the 
length  of  PN  measured  in  the  same  unit  of  length  in  which  u  and 
v  are  measured  is  equal  to  the  tangent  of  the  angle  of  the  ascent. 
Let  us  call  the  line  PN  the  gradient  of  the  function  <p(u,  v)  at  the 
point  u,  v.  The  direction  of  the  gradient  is  perpendicular  to  the 
curve  <p(u,  v)  =  const,  that  passes  through  the  point  u,  v;  for  in 
the  direction  of  the  curve  we  have 

—  =  0 

and  therefore 

a  -  X  =  ±  90°. 

If  PN'  is  the  gradient  of  the  function  $(u,  v)  at  the  point  u,  v,  the 
angle  between  PN  and  PN'  must  be  equal  to  the  angle  formed 


68  GRAPHICAL   METHODS. 

by  the  curves  x  =  const,  and  y  =  const,  that  intersect  at  the 
point  u,  v,  or  equal  to  its  supplement  according  to  the  angle  of 
intersection  that  we  consider. 

Suppose  the  gradients  PN  and  PN'  do  not  vanish  in  any  of 
the  points  in  the  region  of  the  uv  plane  that  we  consider  and 
that  their  length  and  direction  vary  as  continuous  functions  of 
u  and  v.  Let  us  further  suppose  that  the  gradient  PN'  (com- 
ponents: d\f//du,  d^/dfl)  is  for  the  whole  region  on  the  left  side 
of  the  gradient  PN  (components:  d<p/du,  d<p/dv),  or  else  for  the 
whole  region  on  the  right  side  of  the  gradient  PN,  then  it  fol- 
lows that  any  one  of  the  curves  x  =  const,  and  any  one  of  the 
curves  y  =  const,  can  only  intersect  once  in  the  region  considered. 

This  may  be  shown  by  considering  the  directions  of  the  curves 
x  ==•  const,  and  y  =  const,  in  the  uv  plane.  Let  us  consider 
that  direction  on  the  curve  y  =  const,  in  which  x  increases.  If 
this  direction  deviates  from  PN  the  deviation  must  be  less  than 
90°,  because  dx/ds  and  therefore  cos  (a  —  X)  is  positive.  Let  us 
further  consider  that  direction  on  the  curve  x  —  const,  in  which 
y  increases.  If  it  deviates  from  the  direction  of  PN'  the  devia- 
tion must  be  less  than  90°.  Let  us  call  these  directions  the 
direction  of  x  (on  the  curve  y  =  const.)  and  the  direction  of  y 
(on  the  curve  x  =  const.).  Now  if  the  gradient  PN'  is  on  the 
left  of  the  gradient  PN  the  y  direction  must  also  be  on  the  left 
of  PN  (for  if  it  were  on  the  right  of  PN  being  perpendicular  to 
PN  it  would  form  an  obtuse  angle  with  PN')  and  therefore  it 
must  be  on  the  left  of  the  x  direction  (for  if  it  were  on  the  right, 
PN'  being  perpendicular  to  the  x  direction  would  form  an  obtuse 
angle  with  the  y  direction,  which  we  have  seen  to  be  impossible). 
Similarly  it  may  be  seen,  that  if  PN'  is  on  the  right  of  PN,  the 
direction  of  y  will  also  be  on  the  right  of  the  direction  of  x.  If 
therefore  PN'  is  on  the  same  side  of  PN  in  the  whole  region 
considered,  the  direction  of  y  will  also  be  on  the  same  side  of  the 
direction  of  x  for  the  whole  region  considered.  This  excludes 
the  intersection  of  two  curves  x  =  const,  and  y  —  const,  in  more 
than  one  point.  For,  suppose  there  are  two  points  of  inter- 


GRAPHICAL   REPRESENTATION   OF   FUNCTIONS.  69 

section  and  we  pass  along  the  curve  y  =  const,  in  the  direction  of 
x.  At  the  first  point  of  intersection  we  pass  over  the  curve 
x  =  const,  from  the  side  of  smaller  values  of  x  to  the  side  of 
larger  values  of  x.  Now  if  the  values  of  x  go  on  increasing 
as  we  go  along  the  curve  y  =  const,  we  evidently  cannot  get 
back  to  a  curve  x  =  const,  corresponding  to  a  smaller  value  of  x. 
The  only  possibility  of  a  second  point  of  intersection  would  be 
that  the  direction  in  which  the  value  of  x  increases  on  the  curve 
y  =  const,  becomes  the  opposite,  so  that  in  advancing  in  the 
same  direction  in  which  we  came  x  would  decrease  again. 

The  same  holds  for  the  curve 
x=  const.  If  we  pass  from  one 
point  of  intersection  with  a 
curve  y  =  const,  along  a  curve 
x  =  const,  to  a  second  point 
of  intersection  with  the  same 
curve  the  only  possibility  is 
that  the  direction  of  y  also  be- 
comes opposite.  This  is  ex- 
cluded as  in  contradiction  with  FlQ>  49> 
the  direction  of  y  being  on  the 
same  side  of  the  direction  of  x  throughout  the  whole  region  (Fig.49) 

It  will  be  useful  to  look  at  it  from  another  point  of  view.  Let 
us  consider  a  point  A  in  the  uv  plane  corresponding  to  the 
values  u,  v  and  let  us  increase  u  and  v  by  infinitely  small  positive 
amounts  du  and  dv,  so  that  we  get  four  points  ABCD,  forming  a 
rectangle  corresponding  to  the  coordinates. 

A  :  u,  v,  B  :  u  +  du,  v;  C  :  u,  v  -f-  dv;  D  :  u  -f-  du,  v  +  dv. 

In  the  xy  plane  these  points  are  depicted  in  the  points  A, 
B,  C,  D,  the  intersections  of  two  curves  u  and  u  +  du  with  two 
curves  v  and  v  +  dv  (Fig.  50). 

The  projections  of  the  line  AB  in  the  xy  plane  on  the  axes  of 
coordinates  are  obtained  by  calculating  the  changes  of  x  and  y 
for  a  constant  value  of  v  and  a  change  du  in  the  value  of  u 


70 


GRAPHICAL  METHODS. 


Similarly  the  projections  of  AC  are  obtained  by  calculating  the 
changes  of  x  and  y  for  a  constant  value  of  u  and  a  change  dv  in  the 
value  of  V 

,          d<p.        .        d#, 
<fe=  -^dv,     dy2=~dv. 

Denoting  the  lengths  of  AB  and  AC  by  d$i  and  dsz  and  the  angles 
that  the  directions  of  AB  and  AC  form  with  the  direction  of  the 


dv 


FIG.  50. 


positive  axis  of  x  (the  angles  counted  in  the  usual  way)  by  71 
and  72  we  have: 

dxi  =  dsi  cos  71,    dyi  =  dsi  sin  71 


and 
or 

and 

We  may  call 


=  ds2  cos  72,     dyz  =  dsz  sin  72, 


d<p 


the  scale  of  depiction  at  A  in  the  direction  AB  and 


GRAPHICAL   REPRESENTATION   OF  FUNCTIONS.  71 


dv 

the  scale  of  depiction  at  A  in  the  direction  AC.  It  is  here  under- 
stood that  the  uv  plane  is  the  original,  which  is  depicted  on  the 
xy  plane.  If  we  take  it  the  other  way  the  scales  of  depiction 
in  the  directions  AB  and  AC  are  the  reciprocal  values  dulds\ 
and  dvfds-i. 

The  area  of  the  parallelogram  ABCD  in  the  xy  plane  is 


•     /  \ 

an  fe  -  70  -  -  £•  ^ 

According  to  the  way  in  which  the  angles  72  and  71  are  defined 
sin  (72  ~~  7i)  is  positive,  when  the  direction  AC  points  to  the  left 
of  the  direction  AB  (assuming  the  positive  axis  of  y  to  the  left 
of  the  positive  axis  of  x),  and  sin  (72  —  71)  is  negative,  when  AC 
points  to  the  right.  Now  dudv  is  equal  to  the  area  of  the  rectangle 
ABCD  in  the  uv  plane.  Therefore  the  value  of 


du  dv        dv  du 

is  the  ratio  of  the  areas  ABCD  in  the  two  planes  and  its  positive 
or  negative  sign  denotes  the  relative  position  of  the  directions 
AB  and  AC  in  the  xy  plane.  We  may  call  this  ratio  the  scale 
of  depiction  of  areas  at  the  point  A. 


du  dv        dv  du 

is  called  the  functional  determinant  of  the  functions  p(u,  v)  and 
t(u,  «). 

We  have  found  the  scale  of  depiction  of  lengths  in  the  direc- 
tions AB  and  AC.  Let  us  now  try  to  find  it  in  any  direction 
whatever.  From  any  point  A  in  the  uv  plane,  whose  coordinates 
are  u  and  x,  we  pass  to  a  point  D  close  by  whose  coordinates  are 
u  +  Aw,  x  +  At>.  In  the  xy  plane  we  find  the  corresponding 
points  A  and  D  with  coordinates  (Fig.  51). 


72 


GRAPHICAL  METHODS. 


^ .  x  =  <p(u,  v)     p.  x  +  Ax  =  ??(M  -f-  AM,  u  +  Au) 
y  =  &(u.  c)        '  y  ~r~  Aw  =  ^(M  ~j~  AM,  ??  -|-  A??) 

We  expand  according  to  Taylor's  theorem,  and   writing  for 
shortness 

d  &  3  <p  dw  $\1/ 

we  find 

Ax  =  ^«AM  +  <f>v&v  +  terms  of  higher  order, 

Ay  =  faAu  +  faAv  +  terms  of  higher  order. 


Au 


Fio.  61. 

The  length  of  AD  and  the  angle  of  its  direction  we  denote  by 
Ar  and  a  in  the  uv  plane  and  by  A*  and  X  in  the  xy  plane. 
The  limit  of  the  ratio  A*/Ar,  to  which  it  tends,  when  D  approaches 
A  without  changing  the  direction  AD  is  the  scale  of  depiction 
at  the  point  A  in  the  direction  AD. 
Writing 

AM   =  Ar  cos  a, 
Az>  =  Ar  sin  a, 
we  obtain 

Aa:  =  (<f>u  cos  a  +  <pv  sin  a)Ar  +  terms  of  higher  order, 
Ay  =  (^u  cos  a  +  fa  sin  a)Ar  +  terms  of  higher  order. 

Dividing  by  Ar  and  letting  Ar  decrease  indefinitely,  we  have  in 
the  limit 

dx 

--=  (pu  cos  a  +  V*  sin  a, 


GRAPHICAL  REPRESENTATION   OF   FUNCTIONS.  73 

dy 
j-  =  \f/u  cos  a  +  ^v  sin  a. 

For  dx/dr  and  <ft//dr  we  may  also  write  dsfdr  cos  X,  ds/dr  sin  X. 

& 

T"  cos  X  =  ¥>M  cos  a  +  <£„  sin  a, 

ds  . 

T"  sin  X  =  ^u  cos  a  -{-  \f/v  sin  a. 

These  equations  show  the  scale  of  depiction  ds/dr  corresponding 
to  the  different  directions  X  in  the  x,  y-plane  and  a  in  the  u,  v- 
plane. 

By  introducing  complex  numbers  we  can  show  the  connection 
still  better. 
Let  us  denote 

dx      dy  .      ds  x< 
2  —  -j-  +  -r 1  =  T  e *, 
dr  r  dr        dr 

Zl  =    Vu  +  tui, 
22  =    *»»  +  &0. 

Multiplying  the  second  of  the  two  equations  by  i  and  adding 
both  they  may  be  written  as  one  equation  in  the  complex  form: 

2  =  Zi  cos  a  -f-  Zz  sin  a. 

The  modulus  of  z  is  the  scale  of  depiction  of  the  wo  plane  at  the 
point  A  in  the  direction  a.     The  angle  of  z  gives  the  direction  in 
the  xy  plane  corresponding  to  the  direction  a.     For  a  =  0  we 
have  z  =  Zi  and  for  a  =  90°,  z  =  22. 
Let  us  substitute 


COS  a  — 


2  2t 

and  write 


2        ' 
so  that  the  expression  for  z  becomes 


74  GRAPHICAL   METHODS. 

z  =  aeai  +  be~ai. 

This  suggests  a  simple  geometrical  construction  of  the  complex 
numbers  z  for  different  values  of  a.     The  term  aeai  is  represented 
by  the  points  of  a  circle  described  by  turning  the  line  that 
represents  the  complex  number  a  round 
the  origin  through  the  angles  a=Q-  •  -2ir. 
The   term   be~ai  is  represented    by  the 
points  of  a  circle  described   by  turning 
the  line  that  represents  b  round  the  ori- 
gin in  the  opposite  direction  through  the 
angles  a  =  0  •  •  •  -  2ir  (Fig.  52).     The 
addition  of  the  two  complex  numbers 
jijg  62  ae^i  and  be  ai  for  any  value  of  a  is  easily 

performed.      The   points  corresponding 

to  the  complex  numbers  z  describe  an  ellipse,  whose  two  princi- 
pal axes  bisect  the  angles  between  a  and  b.  This  is  easily  seen 
by  writing 

a  =  ne(a<r-ai)i,     b  =  r2e(ao+ai)i. 

aQ  corresponds  to  the  direction  bisecting  the  angle  between  a 
and  b  and  «i  denotes  half  the  angle  between  a  and  6  (positive  or 
negative  according  to  the  position  of  a  and  6). 


=  fa  +  r2)  cos  (a  —  ai)  +  fa  —  r2)  sin  (a  —  aji. 

Denoting  the  coordinates  of  the  complex  number  ze~aoi  by  £  and  77 
we  have 


=  sin  (a  —  ai), 
and  consequently  the  equation  of  an  ellipse 

fa  +  r2)2  ~*~  fa  -  r2)2  =  1' 


GRAPHICAL   REPRESENTATION   OF  FUNCTIONS. 


75 


This  ellipse  turned  round  the  origin  through  an  angle  equal  to 
«o  gives  us  the  points  corresponding  to  z.  The  principal  axes 
are  2(n  +  r2)  and  2(n  -  r2)  (Fig.  53).  The  construction  of 


FIG.  53. 

Fig.  53  is  obvious.  After  plotting  z\  and  Zz  we  find  Zz/i  and 
—  Zz/i  by  turning  AZ2  through  a  right  angle  to  the  right  and  to 
the  left.  From  these  points  lines  are  drawn  to  Z\.  The  bisection 
of  these  lines  give  a  and  b. 

The  figure  shows  that  in  case  a  and  6  have  the  same  modulus, 
the  triangle  —  Zz/i,  Z\,  Zz/i  becomes  equilateral  and  AZ\  is  per- 
pendicular to  the  line  joining  —  Zz/i  and  Zj/t.  In  this  case  AZ\ 
and  AZz  would  have  the  same  or  the  opposite  direction.  But  as 
zi  =  <f>u  +  tui,  Zz  =  <f>v  +  iM,  this  would  mean  that  (f>u\f/v  —  <pv^u 
=  0. 

The  radii  of  the  ellipse  (Fig.  53)  measured  in  the  unit  used 
give  the  different  scales  of  depiction  corresponding  to  the  dif- 
ferent directions  in  the  xy  plane.  We  might  also  say  the  ellipse 
is  the  image  in  the  xy  plane  of  an  infinitely  small  circle  in  the 
uv  plane,  magnified  in  the  proportion  of  the  infinitely  small  radius 
to  1,  with  its  center  in  A. 

Zi  corresponds  to  a  =  0  and  Z2  to  a  =  90°  and  for  a  =  0  to  90° 


76 


GRAPHICAL   METHODS. 


Z  moves  on  the  ellipse  from  Z\  to  Zz  through  the  shorter  way. 
—  Zi  corresponds  to  a  =  180°  and  —  Zz  to  a  =  270°.  Now  we 
have  shown  above  that  a  positive  value  of  the  functional  deter- 
minant put*  —  Vvtu  means  that  Zz  is  on  the  positive  side  of  Z\, 
so  that  in  this  case  Z  moves  in  the  positive  sense  (that  is,  in  the 
direction  from  the  positive  axis  of  x  to  the  positive  axis  of  y)  with 
increasing  values  of  a.  With  a  negative  value  Z  moves  in  the 
opposite  direction. 

Let  us  now  suppose  that  the  curves  x  =  const,  and  y  =  const,  in 
the  uv  plane  intersect  except  on  a  certain  curve  where  their  direc- 


-y, 

-1/4 


u 
FIG.  54. 

tions  coincide  in  the  way  shown  in  Fig.  54.  On  this  curve  the 
functional  determinant  D  =  ipu^v  —  <pv^u  must  vanish  because 
the  directions  of  the  gradients  coincide.  Let  us  see  what  the 
depiction  on  the  xy  plane  is  like. 

Running  along  one  of  the  curves  y  =  const.,  say  y  =  y\, 
toward  the  curve  D  =  0  we  intersect  the  curves  x  =  x±,  x$,  Xz 
until  at  the  point  A  on  the  curve  x  =  xtwe  reach  the  curve  D  =  0. 
In  the  xy  plane  the  corresponding  path  is  a  parallel  to  the  axis 
of  x  at  a  distance  y\  passing  through  #4,  z3,  Xz  and  reaching  a 
point  A  at  XL  If  we  now  proceed  on  the  curve  y  =  yi  in  the 
uv  plane  beyond  the  curve  D  =  0,  we  again  intersect  the  curves 
Xz,  x3,  etc.,  but  in  the  inverse  order.  Thus  the  corresponding 
path  in  the  xy  plane  does  not  pass  beyond  A,  but  turns  back 


GRAPHICAL   REPRESENTATION   OF   FUNCTIONS.  77 

through  the  same  points  z2,  y\\  x3,  y1}  etc.  The  same  holds  for 
any  of  the  other  lines  y  =  const.  If  we  trace  the  line  in  the 
xy  plane  that  corresponds  to  the  points  in  the  uv  plane,  where 
the  curves  x  =  const,  and  y  =  const,  touch,  we  find  the  depiction 
of  the  uv  plane  only  on  one  side  of  the  curve  in  the  xy  plane. 
The  other  side  has  no  corresponding  points  u,  v.  However  to 
every  point  C  on  this  side  of  the  curve,  there  are  two  correspond- 
ing points  C  in  the  uv  plane,  one  on  either  side  of  the  curve 
D  =  0.  Imagine  two  sheets  of  paper  laid  on  the  xy  plane;  let 
them  both  be  cut  along  the  curve  AB.  Retain  only  the  two 
pieces  on  this  side  of  the  curve  and  paste  them  together  along 
the  curve.  The  uv  plane  is  in  this  way  depicted  on  the  paper 
in  such  a  way  that  there  is  one  point  and  one  only  on  the  paper 
corresponding  to  each  point  in 
the  region  of  the  uv  plane  con- 
sidered. The  curve  D  =  0  in 
the  uv  plane  corresponds  to  the 
rim  where  the  two  pieces  of  pa- 
per are  pasted  together.  Any 
line  straight  or  curved  passing  FIG.  55. 

over  the  curve  D  =  0  in  the  uv 

plane,corresponds  to  a  line  running  from  one  of  the  sheets  onto  the 
other.  It  need  not  change  its  direction  abruptly  when  it  reaches 
the  rim  and  passes  onto  the  other  sheet.  For  it  may  touch  the 
rim  in  the  direction  of  its  tangent.  This  is  actually  the  rule 
and  the  abrupt  change  of  direction  is  the  exception.  Any  line 
LAL  (Fig.  55)  in  the  uv  plane,  whose  tangent  as  it  crosses  the 
curve  D  =  0  at  A  does  not  coincide  with  the  common  tangent 
of  the  curves  x  =  const,  and  y  =  const,  will  correspond  to  a  line 
in  the  xy  plane,  that  does  not  change  its  direction  abruptly 
when  it  touches  the  rim. 

This  is  best  understood  analytically.  Let  us  consider  corre- 
sponding directions  at  the  points  A  in  the  uv  plane  and  in  the 
xy  plane.  We  have  seen  above  that  corresponding  directions 
(Fig.  56)  are  connected  by  the  equations 


78 


GRAPHICAL  METHODS. 


dy/dr- 


dx/dr 


FIG.  56. 


<&      dx 

cos  X  y~  =  -j"  =  <f>u  cos  a  +  <f>v  sin  a, 

ds      dy 
-  =  -7-  =  ^u  cos  a  +  ^c  sm  a. 


At  the  point  A  we  have 


Assuming  that  the  gradients  at  A  do  not  vanish,  so  that  we 
can  write 

<pu  =  r  cos  7  ,     <pv  =  r  sin  7, 

\f/u  =  r'  cos  7',     \l/v  =  r'  sin  7', 

where  r  and  r'  are  positive  quantities,  the  equation  (pu^v—  v»^u=0 
reduces  to  sin  (7  —  7')  =  0,  that  is,  7  =  7'  or  7  =  7'+  180°. 
It  follows  therefore  that: 

ds 
cosX^  =  r  cos  (a  —  7), 

sin  XT"  =  r'  cos  (a  —  7')  =  =*=  r'  cos  (a  —  7). 

Consequently  for  all  directions  a.  in  the  uv  plane  for  which 
cos  (a  —  7)  is  not  zero,  we  have 


tgX 


GRAPHICAL   REPRESENTATION   OF   FUNCTIONS.  79 

That  is  to  say,  we  have  in  the  xy  plane  only  one  fixed  direction 
X  and  the  opposite  corresponding  to  all  the  different  directions 
a  except  only  a  direction  for  which  cos  (a  —  7)  =  0.  In  the 
latter  case,  that  is,  when  the  direction  a  is  perpendicular  to  the 
direction  7  of  the  gradient,  i.  e.,  in  the  direction  of  the  curves 
x  =  const,  and  y  =  const.,  we  have 

ds 


ds 

sin  X  -j-  =  0. 
dr 

Therefore  ds/dr  =  0  and  X  remains  indeterminate.  Any  direction 
X  for  which  tg  X  differs  from  +  r'/r  corresponds  to  a  fixed  direction 
a  =  y  +  90°  or  a  =  y  -  90°,  while  ds/dr  =  0. 

As  the  curve  D  =  0  is  depicted  on  the  rim  of  the  two  sheets 
of  paper,  all  those  lines  that  intersect  the  curve  D  =  0  in  a 
direction  different  from  the  direction  of  the  curves  x  =  const. 
and  y  =  const,  are  depicted  in  the  xy  plane  as  curves  having 
their  tangent  at  A  in  common  with  the  rim.  All  lines  in  one  of 
the  sheets  of  paper  that  touch  the  rim  at  A  in  a  direction  differ- 
ent from  that  of  the  rim  must  be  the  depiction  of  lines  in  the  wo 
plane  that  reach  A  in  the  direction  of  the  lines  x  =  const,  and 
y  =  const.  The  scale  of  depiction  is  zero  in  the  direction  of  the 
curves  x  =  const,  and  y  =  const.  In  any  other  direction  a 
we  find  it  different  from  zero  for: 


=  V(?  +  r'2)  cos2  (a  -  T). 

It  is  a  maximum  in  the  direction  a  =  y  or  y  +  180°  perpendicular 
to  the  curves  x  =  const,  and  y  =  const. 

It  may  help  to  understand  all  these  details  if  we  discuss  an 
example  where  the  depiction  of  the  uv  plane  on  the  xy  plane 
has  a  simple  geometrical  meaning,  the  planes  being  ground  plan 
and  elevation  of  a  curved  surface  in  space.  The  rim  in  the 
xy  plane  is  the  outline  of  the  surface,  the  projection  of  those 


80 


GRAPHICAL  METHODS. 


points  where  the  tangential  plane  is  perpendicular  to  the  plane 

of  elevation. 

Suppose  a  cylinder  of  circular  section  cut  in  two  half  cylinders 

by  a  plane  through  its  axis.     Suppose  one  of  the  half  cylinders 

in  such  a  position  that  its  axis 
forms  an  angle  8  with  the 
ground  plan,  the  plan  of  ele- 
vation being  parallel  to  its 
axis,  Fig.  57.  Let  us  intro- 
duce rectangular  coordinates 
u,  v  in  the  ground  plan  and 
rectangular  coordinates  x,  y 
in  the  plan  of  elevation.  A 
point  P  on  the  cylinder  is  de- 
fined by  certain  values  u,  v 
which  define  its  ground  plan 
and  certain  values  x,  y  which 
define  its  elevation.  It  is 
easily  seen  from  Fig.  57  that 
we  have 


AB 

V 

A 

R 

X 

D 

.1 

P 

4  l 

u 

( 

\l    \ 

and 


where  a  is  the  radius  of  the  section.  Now  let  us  consider  the 
elevation  of  the  points  P  as  a  depiction  of  their  ground  plan. 
The  functions  <p(u,  v)  and  \f/(u, «)  in  this  case  are 

<p(u,  v)  =  u, 
\fs(u,  v)  =  u  tg  d  + 

and 

» 


1 
cos  6 


=  tgS, 


cos  5 I/a2  — 


cos  5  I/a2  — 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 


81 


CD 


FIG.  58. 


The  functional  determinant  vanishes  for  v  =  0  on  the  line  EF. 
The  lines  y  =  const,  are  the  intersections  of  the  cylinder  with 
horizontal  planes.  In  the  plan  of 
elevation  they  are  straight  hori- 
zontal lines;  in  the  ground  plan 
they  are  ellipses  (Fig.  58).  As  we 
pass  along  one  of  these  curves  we 
cross  the  line  EF  in  the  ground 
plan  but  we  only  touch  it  in  the 
plan  of  elevation,  retracing  the  hori- 
zontal line  back  again.  The  lines 
x  =  const,  are  straight  lines  in  both 
planes,  but  in  space  they  corre- 
spond to  ellipses.  Again  as  we 
cross  EF  in  the  ground  plan  we 
only  touch  it  in  the  plan  of  eleva- 
tion and  retrace  the  vertical  line  down  again.  Any  curve  on 
the  cylinder  that  crosses  EF  in  a  direction  not  perpendicular  to 
the  plan  of  elevation  is  projected  in  the  plan  of  elevation  with 
EF  as  its  tangent.  For  the  real  tangent  in  space  lying  in  the 
tangential  plane  of  the  cylinder  can  have  no  other  projection,  if 
not  perpendicular  to  the  plan  of  elevation.  In  this  latter  case 
the  projection  of  the  tangent  is  a  point 
and  the  tangent  of  the  elevation  is  deter- 
mined by  the  inclination  of  the  osculatory 
plane. 

There  is  a  particular  case  to  be  consid- 
ered, when  the  curve  D  =  0  in  the  uv  plane 
coincides  with  one  of  the  curves  x  =  const, 
or  y  =  const.  (Fig.  59),  assuming  the  gra- 
dients of  the  functions  <p(u,  v)  and  ^(u,  v) 
not  to  vanish  at  the  points  of  this  curve.  We  have  seen  that  at 
a  point  where  D  =  0  the  scale  of  depiction  must  vanish  in  the 
directions  of  the  curve  x  =  const,  or  y  =  const.  Let  the  curve 
D  =  0  coincide  with  a  line  x  =  const.,  then  it  follows  that  the 

7 


D=o 


FIG.  59. 


82 


GRAPHICAL   METHODS. 


length  of  the  depiction  of  this  curve  is  zero  and  the  depiction 
must  be  contracted  in  a  point.  For  the  length  of  the  depiction 
of  a  curve  x  =  const,  is  given  by  an  integral 


where  dr  denotes  an  element  of  the  curve  and  ds/dr  the  scale 
of  depiction  in  the  direction  of  the  curve.  As  ds/dr  is  zero  all 
along  the  curve  the  integral  must  necessarily  vanish. 

As  an  example  let  us  con- 
sider 

X  =  UV, 

y  =  v. 

The  lines  x  =  const,  in  the  uv 
*      plane  are   equilateral   hyper- 
bolas, the  lines  y  =  const,  are 
parallels  to  the  axis  of  u  (Fig. 
60).     Along  the  axis  of  u  we 
have  at  the  same  time  y  —  0, 
x=  0    and  D=  v  =  0.      The 
whole  axis  of  u  is  depicted  in 
the  point  x  =  0,  y  —  0  of  the  xy  plane. 

Let  us  finally  consider  the  case  where  the  scale  of  depiction 
at  any  point  is  the  same  in  all  directions,  though  it  need  not  be 
the  same  at  different  points. 
Writing  as  before 


FIG. 


dx  ,   dy  .      ds  .. 

3  =  T  +  :r*  =  Te  » 
dr       dr         dr 

the  connection  between  the  scale  of  depiction  ds/dr  and  the 
angles  X,  a  determining  corresponding  directions  in  the  xy  plane 
and  in  the  uv  plane  is  given  by  the  equation 

Z  =  z\  cos  a  +  Zz  sin  a, 


GRAPHICAL   REPRESENTATION   OF  FUNCTIONS.  83 


or 
where 


In  the  case  where  the  scale  of  depiction  ds/dr,  that  is  to  say,  the 
modulus  of  z,  is  independent  of  a,  one  of  the  constants  a  or  b 
must  vanish,  as  we  see  at  once  from  the  construction  of  z  (Fig. 
52).  Let  us  consider  the  case  6  =  0, 

,-««--£.". 

dr 

The  complex  number  a  may  be  written  \  a  \  e^,  where  |  a  \ 
denotes  the  modulus  of  a  and  a0  the  angle.     Both  may  vary 
from  point  to  point,  but  at  every  point  they  have  fixed  values. 
Consequently  we  have 

-j£  =  |  a  |     and     X  =  a  +  «0. 

That  is  to  say,  from  an  angle  a  determining  a  direction  in  the 
uv  plane,  we  find  the  angle  X  determining  the  corresponding 
direction  in  the  xy  plane  by  the  addition  of  a  fixed  value  a0- 
Any  two  directions  a,  a'  will  therefore  form  the  same  angle  as 
the  corresponding  directions  X,  X'  in  the  xy  plane.  The  same  is 
true  when  a  =  0  and  z  =  6e~a<.  The  only  difference  is  that  in 
this  latter  case  the  direction  of  z  rotates  in  the  opposite  sense 
with  increasing  values  of  a. 

Analytically  depictions  of  this  kind  are  represented  by  func- 
tions of  complex  numbers, 

x  +  yi  =  f(u  +  m}     or    x  +  yi  =  f(u  —  m). 

Assuming  the  function  to  possess  a  differential  coefficient  we  have 

dx       dy.  ^ 


fr 


84 


GRAPHICAL   METHODS. 


and  therefore  either 


Hence  in  the  first  case 
a  =  |(zi  +  Z 
and  in  the  second  case 


or 


a  =  0,    b  =  ZL 


§  11.  Other  Methods  of  Representing  Relations  between  Three 
Variables.  —  The  depiction  of  one  plane  on  another  may  be  used 
to  generalize  the  graphical  representation  of  a  function  of  two 
variables  or  a  relation  between  three  variables,  as  we  prefer 
to  say. 

As  we  have  seen  before,  an  equation 

g(x,  y,  z)  =  0 

between  three  variables  x,  y,  z  can  be  represented  by  taking  x 
and  y  as  rectangular  coordinates  and  plotting  the  curves  z  = 
const.  (Fig.  61)  for  equidistant  val- 
ues of  z.  Suppose  now  the  xy  plane 
to  be  depicted  on  another  plane. 
The  lines  x  =  const.,  y  =  const,  and 
z  =  const,  will  be  represented  by 
three  sets  of  curves.  The  fact  that 
three  values  x,  y,  z  satisfy  the  equa- 
tion g(x,  y,  z)  =  0  is  shown  geo- 
metrically by  the  intersection  of 
the  three  corresponding  curves  in 
one  point. 

Another  method  for  representing  certain  relations  between 
three  variables  u,  v,  w  consists  in  drawing  three  curves,  each 
curve  carrying  a  scale.  The  values  of  u,  v,  w  are  read  each  on 
one  of  the  three  scales.  The  relation  between  three  values  u,  v, 
w  is  represented  geometrically  by  the  condition  that  the  corre- 
sponding points  lie  on  a  straight  line  (Fig.  62).  This  method  is 


FIG.  61. 


GRAPHICAL   REPRESENTATION   OF   FUNCTIONS.  85 

far  more  convenient  than  the  one  using  three  sets  of  curves.  It  is 
less  trouble  to  place  a  ruler  over  two  points  u,  v  of  two  curves 
and  read  the  value  w  on  the  scale  of  the  third  than  to  find  the 
intersection  of  two  curves  u  =  const,  and  v  =  const,  among  sets 
of  others,  pick  out  the  curve  w  =  const,  that  passes  through  the 


same  point  and  read  the  value  of  w  corresponding  to  it.  For  we 
must  consider  that  the  curves  corresponding  to  certain  values 
of  u  and  v  are  generally  not  drawn,  but  must  be  interpolated  and 
so  must  the  curve  w  =  const.  It  is  true  that  interpolations  are 
necessary  with  both  methods,  but  the  interpolation  on  scales 
like  those  in  Fig.  62  is  easily  done. 

It  must  however  be  understood  that  while  the  three  sets  of 
curves  form  a  perfectly  general  method  for  representing  any  rela- 
tion between  three  variables,  the  other  method  is  restricted  to  cer- 
tain cases.  In  order  to  investigate  this  subject  more  fully  we 
shall  have  to  explain  the  use  of  line  coordinates. 

When  we  apply  rectangular  coordinates  x,  y  to  define  a  certain 
point  in  a  plane,  we  may  say  that  x  determines  one  of  a  set  of 
straight  lines  (parallel  to  the  axis  of  ordinates)  and  y  determines 
one  of  another  set  of  straight  lines  (parallel  to  the  axis  of  abscissas) 
and  the  point  is  the  intersection  of  the  two  (Fig.  63,  /).  A 
similar  method  may  be  used  to  determine  a  certain  straight  line 
in  a  plane.  Let  x  determine  a  point  on  a  certain  straight  line, 
x  being  its  distance  from  a  fixed  point  A  on  the  line  measured 
in  a  certain  unit  and  counted  positive  on  one  side  and  negative 
on  the  other.  Let  y  define  a  point  on  another  straight  line 


86 


GRAPHICAL   METHODS. 


parallel  to  the  first,  y  being  its  distance  from  a  fixed  point  B  on 
the  line  measured  in  the  same  way  as  x.  The  straight  line 
passing  through  the  two  points  is  thus  determined  by  the  values 


<D  y 


(XI)  .a 


FIG.  63. 

x  and  y  and  for  all  possible  values  of  x  and  y  we  obtain  all  the 
straight  lines  of  the  plane  except  those  parallel  to  the  lines  on 
which  x  and  y  are  measured.  For  simplicity  we  choose  AB 
perpendicular  to  the  two  lines  (Fig.  63,  77).  Let  us  call  x  and  y 
the  line  coordinates  of  the  line  connecting  the  two  points  x  and 
y  in  Fig.  63,  77,  in  the  same  way  as  x  and  y  in  Fig.  63,  7,  are 
called  the  point  coordinates  of  the  point  where  the  two  lines 
x  and  y  intersect. 

A  linear  equation  between  point  coordinates 

y  =  mx  +  n 

is  the  equation  of  a  straight  line.  That  is  to  say,  all  the  points 
whose  coordinates  satisfy  the  equation  lie  on  a  certain  straight 
line.  If,  on  the  other  hand,  we  regard  x  and  y  as  line  coordinates 
we  find  the  analogous  theorem:  all  the  straight  lines  whose 
line  coordinates  satisfy  the  equation 

y  =  mx  +  ju 

pass  through  a  certain  point.  The  equation  is  therefore  called 
the  equation  of  the  point. 

In  order  to  show  this  let  us  first  draw  the  line  x  =  0,  y  =  n 
(APO  in  Fig.  64).     If  now  for  any  value  of  x  we  make  AR  =  x 


GRAPHICAL  REPRESENTATION"  OF  FUNCTIONS. 


87 


and  PQ  =  mx,  the  point  of  intersection  of  RQ  and  AP  must  be 
independent  of  x,  for 


PO 

Af\ 

AO 


mx 

- 

x 


The  ratio  PO/AO  determines  the  position  of  0  and  as  it  is 
independent  of  x  and  the  positions  of  A  and  P  are  also  inde- 
pendent of  x,  the  same  is  true  for  0. 
For  negative  values  of  m,  PO  and 
AO  have  opposite  directions  so  that 
0  lies  between  A  and  P. 

For  a  given  point  0,  we  can  find 
the  corresponding  values  of  m  and  M 
by  joining  0  with  the  points  A  and 
the  point  corresponding  to  x  =  1. 
If  P  and  Q  are  the  intersections  of 
these  lines  with  the  line  on  which  y 
is  measured,  we  have  BP  =  M  and  PQ  =  m.  Any  point  in  the 
plane  thus  leads  to  an  equation 

y  =  mx  +  n, 

except  the  points  on  the  line  on  which  x  is  measured.     For 
m  =  0  the  equation  reduces  to 


FIG.  64. 


that  is,  the  equation  of  a  point  on  the  line  on  which  y  is  measured. 

Instead  of  y  =  mx  +  /z,  we  might  also  write  x  =  m'y  +  /*', 
and  go  through  similar  considerations  changing  the  parts  of  x 
and  y.  This  form  does  not  include  the  points  on  the  line  on 
which  y  is  measured,  but  it  does  include  the  points  on  the  line 
on  which  x  is  measured.  For  these  we  have  m'  =  0. 

The  general  equation  of  a  point  in  line  coordinates  is  given  in 
the  form 

ax  +  by  +  c  =  0, 

from  which  we  may  derive  either  of  the  first-mentioned  forms 
dividing  it  by  a  or  6. 


GRAPHICAL   METHODS. 

Dividing  by  c  another  convenient  form  is  obtained, 


—  c       —  c 
or  writing 

—  c  _  —  c 

a  °*       6 


, 

ZO          2/0 

x0  determining  the  point  of  intersection  of  the  line  BO  (Fig.  64) 
and  the  a>line,  while  3/0  determines  the  point  of  intersection  of 
the  line  AO  with  the  y-line. 

A  curve  may  be  given  by  an  equation 

ai(u}x  +  bi(u)y  +  ci(u)  =  0, 

in  which  ai(u),  bi(u},  c\(u)  are  functions  of  a  variable  u.  Any 
value  of  u  furnishes  the  equation  of  a  certain  point  and  as  u 
changes  the  point  describes  the  curve.  Let  us  suppose  the  curve 
drawn  and  a  scale  marked  on  it  giving  the  values  of  u  in  certain 
intervals  sufficiently  close  to  interpolate  the  values  of  u  be- 
tween them.  Two  other  curves  are  in  the  same  way  given  by 
the  equations 

02(20*  +  bz(v)y  +  <%(*)  =  0, 

(w)  =  0, 


and  scales  on  these  curves  mark  the  values  of  v  and  w. 

Now  we  are  enabled  to  formulate  the  condition  which  must  be 
satisfied  by  the  values  u,  v,  w  in  order  that  the  three  corresponding 
points  lie  in  one  straight  line.  If  x  and  y  are  the  line  coordinates 
of  the  line  passing  through  the  three  points,  x  and  y  must  satisfy 
all  three  equations  simultaneously. 

Consequently  the  determinant  of  the  three  equations  must 
vanish 


&iC3)  +  as(biC2  —  6ad)  =  0, 
and,  vice  versa,  if  the  equation  between  u,  v,  w  may  be  brought 


«» 


GHAPHICAL   REPRESENTATION   OF   FUNCTIONS.  8S 

into  this  form  where  a\,  bi,  c\  are  any  functions  of  u,  02,  b%,  GZ  any 
functions  of  v  and  a$,  63,  c$  any  functions  of  w,  we  can  form  the 
equations 

a\x+  hy  +  ci  =  0, 

d2X  +  Ihy  +  cz  =  0, 
+  c3  =  0, 


and  represent  them  graphically  by  curves  carrying  scales  for 
u,  v,  w.  The  relation  between  u,  v,  w  is  then  equivalent  to  the 
condition  that  the  corresponding  points  on  the  three  curves  lie 
on  a  straight  line.  But  it  must  be  remembered  that  only  a 
restricted  class  of  relations  can  be  brought  into  the  required  form, 
so  that  the  method  cannot  be  applied  to  any  given  relation. 
The  equation  of  a  point 

ax  +  by  -f-  c  =  0 

remains  of  the  same  form,  when  the  units  of  length  are  changed 
for  x  and  y.  If  x'  denotes  the  number  measuring  the  same  length 
as  the  number  x  but  in  another  unit,  the  two  numbers  must  have  a 
constant  ratio  equal  to  the  inverse  ratio  of  the  two  units.  There- 
fore, by  changing  -the  units  independently,  we  have 

x  =  \x',     y  =  py', 
and  the  equation  of  the  point  may  be  written 

d\xf  +  buy'  +  c  =  0, 
or 

a'x'  +  b'y'  +  c  =  0, 

where  a'  =  Xa  and  b'  —  nb. 

It  is  sometimes  convenient  to  define  the  line  coordinates  ir 
another  way.  Let  £  and  77  denote  rectangular  coordinates 
measured  in  the  same  unit,  then  the  equation  of  a  straight  line 
can  be  written 

77  =  tg  <p£  +  770, 

where  <p  is  the  angle  between  the  line  and  the  axis  of  £  and  770 


90  GRAPHICAL   METHODS. 

the  ordinate  of  the  point  of  intersection  with  the  axis  of  ?;. 
Now  let  us  call  tg  <p  and  770  the  line  coordinates  of  the  straight 
line  represented  by  the  equation  and  let  us  denote  them  by  x 
and  y.  Thus  the  values  of  x  and  y  define  a  certain  straight  line 
and  any  straight  line  not  parallel  to  the  axis  of  ordinates  may 
be  defined  in  this  manner.  The  condition  that  a  straight  line 
x,  y  passes  through  a  point  £,  77  is  expressed  by  the  equation 

i]  =  a*  +  y, 
or 

y  =  —  &  +  17- 

If  we  fix  the  values  of  x  and  y,  all  the  values  £,  rj  that  satisfy  this 
equation  represent  the  points  of  the  straight  line  x,  y  and  we 
therefore  call  it  the  equation  of  the  straight  line.  If,  on  the 
other  hand,  we  fix  the  values  of  £  and  77,  all  the  values  x,  y  that 
satisfy  the  equation  represent  the  straight  lines  that  pass  through 
the  given  point  |,  TI,  and  therefore  we  call  it  the  equation  of  the 
point. 

The  more  general  form 

ax  +  by  +  c  =  0 
can  be  reduced  to 


fe- 
lt therefore  represents  the  equation  of  the  point,  whose  rec- 
tangular coordinates   are   £  =  a/6   and  rj  =  —  c/b.     The   case 
where  6  =  0  or 

ax  +  c  =  0 

represents  the  equation  of  a  point  infinitely  far  away  in  the 
direction  <f>  or  the  opposite  direction  <p  -\-  180°,  <p  being  defined  by 


tg*< 

All  the  straight  lines,  whose  coordinates  x,  y  satisfy  the  equation 
ax  +  c  =  0 


GEAPHICAL   REPRESENTATION   OF   FUNCTIONS. 


91 


correspond  to  the  same  value  of  x  but  to  any  value  of  y.  That 
is  to  say,  they  are  all  parallel  and  all  the  straight  lines  of  this 
direction  belong  to  them. 

Let  us  now  discuss  some  of  the  applications  of  line  coordinates 
to  the  graphical  representation  of  relations  between  three 
variables. 

The  relation 


uv  = 


may  be  written  in  the  form 


log  u  +  log  v  =  log  w, 


or 


when 

x  =  log  u 


x  +  y  =  log  w, 


and    y  = 


FIG.  65. 


Let  us  plot  x  and  y  as  line  co- 
ordinates on  two  parallel  lines  (Fig. 
65),  with  scales  for  the  values  of  u 
and  v.  The  equations  x  =  log  u 
and  y  =  log  v  may  be  regarded  as  the  equations  of  the  points  of 
these  two  scales.  The  equation 

x  +  y  =  log  w 

for  any  value  of  w  is  the  equation  of  a  point.  It  can  easily  be 
constructed  as  the  intersection  of  any  lines  x,  y  satisfying  its 
equation.  For  instance,  the  line  x  =  log  w,  y  =  0  and  the  line 
£  =  0,  y  =  log  w.  The  first  line  is  found  by  connecting  the 
scale  division  u  =  w  of  the  w-scale  with  the  point  B,  the  second 
by  connecting  the  scale  division  v  =  w  of  the  0-scale  with  the 
point  A.  If  the  units  of  x  and  y  are  taken  of  the  same  length,  the 
point  of  intersection  will  lie  in  the  middle  between  the  two  lines 
carrying  the  u  and  v  scales  on  a  line  parallel  to  the  two  other  lines 
and  the  w-scale  will  be  half  the  size  of  the  other  two  (Fig.  65). 


92 


GRAPHICAL  METHODS. 


The  relation 


log  u  +  log  v  =  log  w 
expresses  the  condition  that  the  three  equations 


x  =  log  u,    y  =  log  t>,    x  +  y  =  log  w 

are  satisfied  simultaneously  by  the  same  values  of  x  and  y,  that 
is  to  say,  that  the  three  points  on  the  u,  v,  w  scales  corresponding 
to  the  values  pf  u,  v,  w  lie  on  the  same  straight  line  x,  y. 
The  more  general  relation 


where  a.  and  /3  are  any  given  values,  can  be  treated  in  the  same 
manner.  Thus  the  pressure  and  volume  of  a  gas  undergoing 
adiabatic  changes  may  be  represented.  In  this  case  we  have 

pvk  =  w, 

where  p  denotes  the  pressure,  v  the  volume  and  k  and  w  con- 
stants. 

For  a  given  gas  k  has  a  given  value,  but  w  depends  on  the 
quantity  of  the  gas  considered. 

We  write 

x  =  log  p,    y  =  log  v. 
The  relation  then  takes  the  form 

x  +  ky  =  log  w, 

and  represents  a  point  which  may  be  con- 
structed by  the  intersection  of  any  two 
i  straight  lines  x,  y,  whose  coordinates  sat- 
isfy the  equation,  for  instance 


and 


x  =  log  w,    y  =  0 

1 


=,    y  = 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 


93 


+0.5 


+1.0 


+0.5 


The  first  line  connects  the  point  B  (Fig.  66)  with  the  scale 
division  p  =  w  of  the  p  scale  and  the  second  line  connects  the 
point  A  with  the  scale  division  of  the  v  scale  for  which  y  =  k  log  w. 
A  perpendicular  from  the  point  of  intersection  on  AB  meets  it  in 
0'  and  as  the  ratio  AO'/O'B  is 
equal  to  the  ratio  of  the  seg- 
ments on  the  p  and  v  scales 
log  w/k  log  w  =  l/k  it  is  inde- 
pendent of  w.  All  the  points 
corresponding  to  different  val- 
ues of  w  lie  on  the  same  par- 
allel to  the  p  and  v  scales  and 
the  w  scale  may  be  obtained 
by  a  central  projection  of  the 
p  scale  on  this  parallel  from 
the  center  B  (Fig.  66).  We 
might  dispense  with  the  con- 
struction of  the  w  scale  as 
long  as  the  straight  line  for 
the  w  scale  is  drawn.  For  in 
using  the  diagram  we  gener- 
ally start  with  values  po,  VQ 
and  want  to  find  other  values 
p,  v,  for  which 

pvk  =  poi)ok.  FlG-  67- 

The  straight  line  connecting  the  scale  divisions  p  and  v  intersects 
the  w  scale  at  the  same  point  as  the  straight  line  connecting  the 
scale  divisions  p0  and  v0,  so  that  we  need  not  know  the  value  of 
poVok.  It  suffices  to  mark  the  point  of  intersection  in  order  to 
find  the  value  of  p,  when  v  is  given  or  the  value  of  v  when  p  is 
given. 

Another  example  is  furnished  by  the  equation 


-1.0 


-1.5 


-0.5 


-LO 


-1.5 


-2.0 


94  GRAPHICAL  METHODS. 

If  we  regard  x  and  y  as  line  coordinates  any  value  of  w  determines 
the  equation  of  a  point.  We  plot  the  curve  formed  by  these 
points  with  a  scale  on  it  indicating  the  corresponding  values  of  w. 
Any  values  of  x  and  y  determine  a  straight  line  whose  inter- 
sections with  the  w  scale  furnish  the  roots  of  the  equation.  Each 
point  of  the  w  scale  may  be  constructed  by  the  intersection  of 
two  straight  lines,  whose  coordinates  x,  y  satisfy  the  equation, 
for  instance 

x  =  0,     y  =  —  it?  .  and    x  =  —  w,     y  =  O.1 

In  Fig.  67  the  w  scale  is  shown  for  the  positive  values  w  =  0  to 
w  =  2.5. 

In  the  same  manner  a  diagram  for  the  solution  of  the  cubic 
equation 

w*  +  xw  +  y  =  0. 

or  of  any  equation  of  the  form 

w*  -f  xw*  +  y  =  0 
may  be  constructed. 

§  12.  Relations  between  Four  Variables. — The  method  can  be 
generalized  for  relations  between  four  variables. 

Suppose  four  variables  u,  v,  w,  t  are  connected  by  the  equation 

g(u,  v,  w,  t)  =  0, 

and  let  us  assume  that  for  any  particular  value  t  =  t0  the  resulting 
relation  between  u,  v,  w  can  be  given  by  a  diagram  of  the  form 
considered  consisting  of  three  curves  carrying  scales  for  u,  v  and 
w.  Let  us  further  suppose  that  for  other  values  of  t  the  scales 
for  u  and  v  remain  the  same,  but  the  scale  for  w  changes.  Then 
we  shall  have  a  set  of  w  scales  corresponding  to  different  values 
of  t.  Connecting  the  points  that  correspond  to  the  same  value 
of  w  we  obtain  a  network  of  curves  t  =  const,  and  w  =  const. 
(Fig.  68).  Any  two  values  u,  v  furnish  a  straight  line  intersecting 
1  For  small  values  of  w,  this  combination  is  not  good  because  the  angle  of 
intersection  is  small.  One  might  substitute  x  =  2,  y  =•  —  V?  —  2w  for  the 
first  line. 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS. 


95 


the  network  of  curves.     The  points  of  intersection  correspond  to 
values  of  t  and  w  that  satisfy  the  given  relation. 
Any  relation  of  the  form 

v(u)f(t,  w)  +  t(v)g(t,  w)  +  h(t,  w)  =  0 

may  be  represented  in  this  way,  <p(u}  denoting  any  function  of 
u,  ^(r)  any  function  of  v  andf(t, 
w),  g(t,  w},  h(t,  w}  any  functions 
of  t  and  w. 

In  this  case  we  need  only  in- 
troduce the  line  coordinates  x, 
y,  writing 

x  =  v(u),    y  =  t(v). 

We  then  obtain  a  linear  equation 
between  x  and  y, 

f(t,  w)x  +  g(t,  w)y  +  h(t,  w)  =  0, 

which  for  any  given  values  of  t 
and  w  represents  the  equation  of 

a  point.  For  a  given  value  of  t  and  variable  values  of  w  we  obtain 
a  curve  t  =  const,  carrying  a  scale  for  w  and  for  a  series  of  values 
of  t  we  obtain  a  set  of  curves  t  =  const.  Similarly  for  a  given 
value  of  w  and  variable  values  of  t  the  equation  furnishes  a  curve 
w  =  const.,  carrying  a  scale  for  t  and  a  series  of  values  of  w 
furnishes  a  set  of  curves  w  =  const.  From  any  given  values 
of  u  and  v  the  line  coordinates  x  and  y  are  calculated  and  the 
points  where  this  straight  line  defined  by  x  and  y  intersects 
the  network  of  the  curves  t  =  const,  and  w  =  const,  furnish 
the  values  t,  w  that  satisfy  the  relation  together  with  the  given 
values  of  u  and  v.  The  relation  between  the  height,  azimuth, 
declination  of  a  celestial  body  and  the  latitude  of  the  point  of 
observation  may  serve  as  an  example.  Let  h,  a,  8  denote  the 
height,  azimuth  and  declination  and  v  the  latitude.  The  angles 
7T/2  —  <p,  7T/2  —  h,  7T/2  —  d  are  the  three  sides  of  a  spherical 


GRAPHICAL  METHODS. 


Fio.  69. 


triangle  PZS  (Fig.  69)  formed  by  the  pole  P,  the  zenith  Z  and  the 
celestial  body  S.     The  azimuth  is  defined  as  the  supplement  of 
the  angle  PZS. 
The  equation  is 

sin  5  =  sin  <p  sin  h  —  cos  <p  cos  h  cos  a. 

We  write 

x  =  cos  a,     y  =  sin  5, 

so  that  the  equation  becomes 

y  =  sin  #>  sin  h  —  x  cos  ^  cos  h. 

We  shall  in  this  case  use  the  second  system  of  line  coordinates 

where  x  is  the  slope  of  the  line  measured  by  the  tangent  of  the 

angle   formed  with  the  axis  of 

abscissas  and  y  is  the  ordinate 

of  the  intersection  with  the  axis 

of  ordinates.     If  £,  ij  denote  the 

rectangular  coordinates   of   the 

point,  the  equation  of  the  points 

takes  the  form 

•n  =  x£  +  y     or     y  =  17  —  £x, 
so  that  in  our  case  we  have 
£  =  cos  <p  cos  h,    i}=  sin  tp  sin  h. 

The  curves  <f>  =  const,  and  h  = 

const,  can  be  drawn  by  means 

of  these  formulas.     It  is  easily 

seen  that  they  are  ellipses  and 

that  the  curves  <p  =  const,  are 

the  same  as  the  curves  h  =  const. 

For  a  definite  value  of  <p  and  a  j^  70 

variable  value  of  h  we  find 


. 

I 


GRAPHICAL  REPRESENTATION   OF   FUNCTIONS.  97 

and  for  a  definite  value  of  h  and  a  variable  value  of  <p 


cos2  h      sin2  h 

Any  of  the  ellipses  intersects  all  the  others  and  in  this  way  they 
form  a  network.  A  point  of  intersection  of  the  ellipse  <p  =  c\ 
and  the  ellipse  h  =  c2  also  corresponds  to  the  values  h  =  c{  and 
<p  =  c2,  as  the  ellipse  <p  =  Ci  is  identical  with  the  ellipse  h  =  c\ 
and  <p  =  GZ  identical  with  h  =  eg  (Fig.  70).  The  easiest  way  to 
find  this  network  consists  in  drawing  the  straight  lines 

£  +  i]  =  cos  (<p  —  k)} 

and  perpendicular  to  them  the  straight  lines 
|  —  77  =  cos  (<p  +  h}, 

for  equidistant  values  of  <f>  +  h  and  <p  —  h.  The  ellipses  run 
diagonally  through  the  rectangular  meshes  formed  by  the  two 
systems  of  straight  lines.  The  scales  for  <p  and  h  are  written 
on  the  axis  of  coordinates,  both  scales  being  available  for  both 
variables.  The  scale  for  5  is  written  on  the  axis  of  ordinates 
and  is  identical  with  the  scale  for  t  and  h  on  this  axis.  For  the 
ordinate  corresponding  to  a  given  value  5  =  c  is  sin  c,  and  this  is 
also  the  ordinate  of  the  point  where  the  ellipse  <p  =  c  or  h  =  c 
intersects  the  axis  of  ordinates.  The  scale  for  the  azimuth  cannot 
be  laid  down  in  exactly  the  same  way  as  that  for  <?,  h  and  5 
because  cos  a  determines  the  slope  of  the  straight  line  x,  y. 
Let  us  draw  a  parallel  to  the  axis  of  ordinates  through  the  point 
£  =  1,  ?/  =  0  and  mark  a  scale  for  the  azimuth  on  it,  making 
77  =  cos  a  (Fig.  70).  A  line  connecting  the  origin  with  any  scale 
division  of  this  scale  has  the  slope  of  the  line  x  =  cos  a,  y  =  sin  5. 
To  bring  it  into  the  position  of  the  line  x,  y  it  must  be  moved 
parallel  to  itself,  until  its  point  of  intersection  with  the  axis  of 
ordinates  coincides  with  the  scale  division  5.  This  suggests 
another  way  of  using  the  diagram.  Let  a  pencil  of  rays  be 
drawn  from  the  origin  to  the  scale  divisions  of  the  azimuth  scale 
(Fig.  70),  and  let  it  be  drawn  on  a  sheet  of  transparent  paper 

8 


98  GRAPHICAL  METHODS. 

placed  over  the  drawing  of  the  ellipses.  For  any  given  value 
of  8  it  is  moved  up  or  down  as  the  case  may  be  so  that  the  center 
of  the  pencil  coincides  with  the  scale  division  5.  As  long  as  the 
celestial  body  does  not  materially  alter  its  declination  the  dia- 
gram in  this  position  will  enable  us  to  find  any  of  the  three 
values  <p,  h,  a  from  the  other  two. 

As  a  second  example  let  us  consider  the  relation  between  the 
declination  8,  the  azimuth  a,  the  hour  angle  t  of  a  celestial  body 
and  the  latitude  p  of  the  point  of  observation. 

The  relation  is  found  by  eliminating  the  height  h  from  the 
equation 

sin  8  =  sin  <p  sin  h  —  cos  <p  cos  h  cos  a. 

For  this  purpose  we  express  sin  h  and  cos  h  by  the  other  angles 
and  substitute  these  expressions  for  sin  h  and  cos  h. 
We  have 

cos  h  =  cos  8  sin  t/sm  a, 

sin  h  =  sin  <f>  sin  8  +  cos  <f>  cos  8  cos  t. 

Substituting  these  values  we  find 

sin  8  =  sin2  <p  sin  8-\-  sin  tp  cos  <p  cos  5  cos  t—  cos  <p  cos  8  sin  t  ctg  a,  or 

cos2  <p  sin  8  =  sin  <p  cos  <p  cos  5  cos  t  —  cos  <p  cos  8  sin  t  ctg  a. 
Dividing  by  cos2  <f>  cos  8  we  finally  obtain 

tg  6  =  tg  <f>  cos  t —  ctg  a. 

COS  <p 

In  order  to  represent  this  relation  graphically  we  introduce  line 
coordinates 

x  =  ctg  a    and     y  =  tg  8 
and  find 

sin  t 

y=tg*cos*-  —  x. 

Let  us  use  the  second  system  of  line  coordinates.  The  rec- 
tangular coordinates  £,  tj  of  the  point  represented  by  the  equation 
are  found  from  it  equal  to : 


GRAPHICAL  REPRESENTATION   OF    FUNCTIONS.  99 

sin  t 


The  curves  <p  =  const,  are  ellipses, 


The  curves  t  =  const,  are  hyperbolas, 


sin2 1       cos2 1 


1. 


The  ellipses  and  hyperbolas  are  confocal,  the  foci  coinciding 
with  the  points  £  =  ±  1,  t\  =  0,  so  that  the  curves  intersect  at 
right  angles. 

The  scale  for  <p  may  be  written  on  the  axis  of  ordinates  at  the 
points  where  it  intersects  the  ellipses.  It  is  identical  with  the 
scale  for  6,  the  ordinate  in  both  cases 
being  the  tangent  of  the  angle  with  the 
only  difference  that  5  is  negative  on 
the  negative  part  of  the  axis  and  <p  is 
not.  The  scale  for  t  may  be  written 
on  one  of  the  ellipses  corresponding  to 
the  largest  value  of  <p  that  is  to  be  taken 
account  of.  This  ellipse  forms  the 
boundary  of  the  diagram,  so  that 
larger  values  of  p  are  not  represented. 
Corresponding  to  the  azimuth  we  draw 
a  pencil  of  rays  on  a  sheet  of  trans- 
parent paper,  which  is  laid  on  the  draw- 
ing of  the  curves.  The  center  of  the 
pencil  is  placed  on  the  scale  division  5 
and  the  azimuth  is  equal  to  the  angles 

that  the  rays  form  with  the  positive  direction  of  the  axis  of  or- 
dinates (Fig.  71).  It  suffices  to  draw  the  curves  and  the  rays 
only  on  one  side  of  the  axis  of  ordinates.  At  the  apex  of  the 


FIG.  71. 


100  GRAPHICAL  METHODS. 

hyperbolas  the  value  of  t  changes  abruptly.  The  line  t  =  Qh  is 
meant  to  start  from  the  focus  £  =  1,  i)  =  0.  When  the  center  of 
the  pencil  of  rays  is  in  the  origin  the  rays  form  the  asymptotic 
lines  of  the  hyperbolas,  a  =  15°  corresponding  to  t  =  lh,  a  =  30° 
to  t  =  2h  and  so  on. 


GIFT  OF 
DR.  GEORGE  F.  McEWEN 


CHAPTER  III. 

THE  GRAPHICAL  METHODS  OF  THE  DIFFERENTIAL  AND 
INTEGRAL  CALCULUS. 

§  13.  Graphical  Integration. — We  have  shown  how  the  ele- 
mentary mathematical  operations  of  adding,  subtracting,  multi- 
plying and  dividing  and  the  inverse  operation  of  finding  the 
root  of  an  equation  can  be  carried  out  by  graphical  methods  and 
how  functions  of  one  or  more  variables  may  be  represented  and 
handled.  But  the  graphical  methods  would  lack  generality  and 
would  be  of  very  limited  use,  if  they  were  not  applicable  to  the 
infinitesimal  operations  of  differentiation  and  integration.  In- 
deed it  is  here  that  they  are  found  of  the  greatest  value.  In 
many  cases,  where  the  calculus  is  applied  to  problems  of  natural 
science  or  of  engineering,  the  functions  concerned  are  given  in  a 
graphical  form.  Their  true  analytical  structure  is  not  known 
and  as  a  rule  an  approximation  by  analytical  expressions  is  not 
easily  calculated  nor  easily  handled.  In  these  cases  it  is  of  vital 
importance  that  the  operations  of  the  calculus  can  be  performed, 
although  the  functions  are  only  given  graphically. 

Let  us  begin  with  integration,  because  it  is  easier  than  differ- 
entiation and  of  more  general  application. 

Suppose  a  function  y  =  f(x)  given  by  a  curve  whose  ordinate  is 
y  and  whose  abscissa  is  x.  The  problem  is  to  find  a  curve,  whose 
ordinate  Y  is  an  integral  of  the  function  f(x), 


Let  us  assume  the  unit  of  length  for  the  abscissas  independent 
of  the  unit  of  length  for  the  ordinates.  The  value  of  Y  measures 
the  area  between  the  ordinates  corresponding  to  a  and  x,  the 
curve  y  =  f(x)  and  the  axis  of  x  in  units  equal  to  the  rectangle 
formed  by  the  units  of  x  and  y. 

101 


102 


GRAPHICAL  METHODS. 


In  the  simple  case  where  f(x)   is  a  constant  the  equation 
=  f(x)  =  c  is  represented  by  a  line  parallel  to  the  axis  of  x  and 


Y  =    I    cdx  =  c(x  —  a). 


7  is  the  ordinate  of  a  straight  line  intersecting  the  axis  of  x  at 
the  point  x  =  a.  The  constant  c  is  the  change  of  Y  for  an 

increase  of  x  equal  to  1. 

v  If  P  is  the  point  on  the 

axis  of  x  for  x  =  —  1  and 
Q  the  point  where  the  line 
y  =  c  intersects  the  axis 
of  ordinates  (Fig.  72)  the 
desired  line  is  parallel  to 

FlQ  72  PQ.     It  is  constructed  by 

drawing  a  parallel  to  PQ 

through  the  point  x  =  a  on  the  axis  of  x  (Fig.  72,  where  a  =  0). 
When  a  given  value  c\  is  added,  so  that  the  equation  becomes 

Y  =  c  (x-  a)  +  ^ 

it  amounts  to  the  same  as  when  the  straight  line  is  moved  in  the 
direction  of  the  axis  of  ordinates  through  a  distance  c\.  For 
x  =  a  we  then  have  Y  =  c\,  so  that  we  obtain  the  line 

Y  =  c(x  -  a)  +  ci, 

by  drawing  a  parallel  to  PQ  through  the  point  x  =  a,  y  =  ci. 

In  the  second  place  let  us  assume  that  the  line  y  =  f(x)  consists 
of  a  number  of  steps,  that  is  to  say,  that  the  function  has  different 
constant  values  in  a  number  of  intervals  x  =  x\  to  Xz,  x%  to  #3, 
etc.,  while  it  changes  its  value  abruptly  at  3%,  x3,  etc.  The 
line  presenting  the  integral 


F  = 


does  not  change  its  ordinate  abruptly.     It  consists  of  a  con- 
tinuous broken  line,  whose  corners  have  the  abscissas  x2,  x3,  etc. 


DIFFERENTIAL  AND   INTEGRAL  CALCULUS. 


103 


The  directions  of  the  different  parts  are  found  in  the  way  just 
described  by  the  pencil  of  rays  from  P  to  the  points  a,  ft,  y,  etc. 
(Fig.  73),  where  the  horizontal  lines  intersect  the  axis  of  ordinates. 


FIG.  73. 

To  construct  the  broken  line  we  draw  a  parallel  to  Pa  through 
the  point  x  =  Xi  (in  Fig.  73  x\  is  equal  to  0)  as  far  as  the  vertical 
x  =  x2.  Through  the  point  of  intersection  with  the  vertical 
x  =  xz  we  draw  a  parallel  to  P/3  as  far  as  the  vertical  x  =  x3. 
Through  the  point  of  intersection  with  the  vertical  x  =  x3  we 
draw  a  parallel  to  Py  and  so  on. 

Finally  let  us  consider  the  case  of  an  arbitrary  function  y  =  f(x) 
represented  by  any  curve.     In  order  to  find  the  curve 


we  substitute  for  y  =  f(x)  a  function  consisting  of  different 
constant  values  in  different  intervals  and  changing  its  value 
abruptly  when  x  passes  from  one  interval  to  the  next,  so  that 
the  line  representing  this  function  consists  of  a  number  of  steps 
leading  up  or  down  according  to  the  increase  or  decrease  of  f(x}. 
These  steps  are  arranged  in  the  following  way.  The  horizontal 


104  GRAPHICAL  METHODS. 

part  AiA2  of  the  first  step  (Fig.  73)  starts  from  any  point  AI 
of  the  given  curve.  The  vertical  part  A2Bi  and  the  following 
horizontal  part  BiB2  are  then  drawn  in  such  a  manner  that  BiB2 
intersects  the  curve  and  that  the  integral  of  the  given  function 
as  far  as  the  point  of  intersection  KI,  is  equal  to  the  integral  of  the 
stepping  line  as  far  as  the  same  point.  That  is  to  say,  the  areas 
between  the  stepping  line  and  the  given  curve  on  both  sides  of 
the  vertical  part  A2B{  have  to  be  equal.  When  Kb  is  fixed  the 
right  position  of  A2Bj.  may  be  found  by  eye  estimate.  The  eye 
is  rather  sensitive  for  differences  of  small  areas.  Besides  a  shift 
of  A2Bi  to  the  right  or  to  the  left  enlarges  one  area  and  diminishes 
the  other  so  that  even  a  slight  deviation  from  the  correct  position 
makes  itself  felt.  In  the  same  way  the  next  step  B2CiC2  is 
drawn  with  its  vertical  part  B2  C\  in  such  a  position  that  the 
areas  on  both  sides  are  equal.  The  integral  of  the  given  curve 
as  far  as  Kc  will  again  have  the  same  value  as  that  of  the  stepping 
line  as  far  as  Kc.  And  so  on  for  the  other  steps.  The  integral 
of  the  stepping  line  is  constructed  in  the  way  shown.  It  is 
represented  by  a  broken  line  beginning  at  the  foot  of  the  ordinate 
of  AI.  The  corners  lie  on  the  vertical  parts  of  the  steps  or 
their  prolongations.  It  is  readily  seen  that  the  broken  line  con- 
sists of  a  series  of  tangents  of  the  integral  curve 

y  = 

and  that  their  points  of  contact  with  the  integral  curve  lie  on 
the  same  verticals  as  the  points  AI,  Kb,  Kc,  etc.  (In  Fig.  73  these 
points  are  denoted  0,  2,  3,  •  •  • .)  That  these  points  lie  on  the 
integral  curve  follows  from  the  arrangement  of  the  steps  which 
make  the  integral  of  the  given  function  at  Kb,  Kc,  •  •  •  equal  to  the 
integral  of  the  stepping  line.  Now  in  the  points  AI,  Kb,  Kc  •  •  • 
the  ordinates  of  the  given  curve  coincide  with  those  of  the 
stepping  line.  Hence  both  integral  lines  must  for  these  abscissas 
have  the  same  direction. 
1  In  Fig.  73  the  lower  limit  is  0. 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS.  105 

Having  constructed  the  broken  line  and  marked  the  points 
2,  3,  4,  •  •  •  (Fig.  73),  the  integral  curve  is  drawn  with  a  curved 
ruler  so  as  to  touch  the  broken  line  in  the  points,  0,  2,  3, 
As  the  given  curve  does  not  change  its  ordinate  abruptly  the 
integral  curve  does  not  change  its  direction  abruptly.  The 
drawing  shows  how  well  the  integral  curve  is  determined  by  the 
broken  line.  There  is  practically  no  choice  in  drawing  it  any 
other  way  without  violating  the  conditions. 

The  ordinate  of  the  integral  curve  is  measured  in  the  same 
unit  as  the  ordinate  of  the  given  curve  y  =  f(x).  It  may  some- 
times be  convenient  to  draw  the  ordinates  of  the  integral  curve 
in  a  scale  different  from  that  of  the  ordinates  of  the  given  curve. 
For  instance  the  value  of  the  integral  may  become  so  large  that 
measured  in  the  same  unit  the  ordinates  of  the  integral  curve 
would  pass  the  boundaries  of  the  drawing  board,  or  else  they  may 
be  so  small  that  their  changes  cannot  be  measured  with  sufficient 
accuracy.  In  the  first  case  the  scale  is  diminished,  in  the  latter 
case  it  is  enlarged.  This  is  done  by  altering  the  position  of  the 
point  P,  the  center  of  the  pencil  of  rays  that  define  the  directions 
of  the  broken  line.  If  P  approaches  0  the  directions  Pa,  P/3,  •  •  • 
become  steeper  to  the  same  degree  as  if  keeping  P  unchanged  we 
had  increased  the  ordinates  of  A\Ai,  B\B%,  •  •  •  in  the  inverse  pro- 
portion of  the  two  distances  PO.  Hence  by  diminishing  the 
distance  PO  the  ordinates  of  the  resulting  broken  line  are  enlarged 
in  the  inverse  proportion.  On  the  other  hand,  by  increasing  the 
distance  PO  the  ordinates  of  the  resulting  broken  line  are  di- 
minished in  the  inverse  proportion  of  the  distances,  because  the 
change  of  the  directions  Pa,  P/3,  •  •  •  caused  by  a  longer  distance 
PO  is  the  same  as  if  the  ordinates  of  AiA2,  BiB2,  •  •  -  were  di- 
minished in  the  inverse  proportion.  The  broken  line  constructed 
by  means  of  the  longer  distance  P'O  will  therefore  be  the  same  as 
if  the  ordinates  of  the  stepping  line  were  diminished.  It  therefore 
leads  to  an  integral  curve  whose  ordinates  are  diminished  in  the 
same  proportion  (Fig.  74). 

The  graphical  integration  of 


106 


GRAPHICAL  METHODS. 


Y  =  £f(x)dz 


is  not  limited  to  values  x  >  a.     The  method  is  just  as  well  applic- 
able to  the  continuation  of  the  integral  curve  for  x  <  a.     The 


E, 


FIG.  74. 


H  I 


steps  have  only  to  be  drawn  from  right  to  left.  The  lower  limit 
a  determines  the  point  where  the  integral  curve  intersects  the 
axis  of  x. 

There  is  a  method  for  the  construction  of  the  vertical  parts 
of  the  steps,  which  may  in  some  cases  be  useful,  though  as  a  rule 
we  may  dispense  with  it  and  fix  their  position  by  estimation. 

Suppose  that  A  and  B  (Fig.  75) 
are  two  points  where  the  curve  is 
intersected  by  the  horizontal  parts 
of  two  consecutive  steps  and  that 
the  curve  between  A  and  B  is  a 
parabola  whose  axis  is  parallel  to 
the  axis  of  x.  The  position  of  the 
vertical  part  of  the  step  between  A 
and  B  can  be  then  found  by  a  simple 

construction.  Through  the  center  C  of  the  chord  AB  (Fig.  75) 
draw  a  parallel  CD  to  the  axis  of  x,  D  being  the  point  of  inter- 
section with  the  parabola.  The  vertical  part  EH  of  the  step  in- 
tersects CD  in  a  point  whose  distance  from  C  is  twice  the  distance 


FIG.  75. 


DIFFERENTIAL   AND    INTEGRAL    CALCULUS 


107 


from  D.  That  this  is  the  right  position  of  EH  is  shown  as  soon 
as  we  can  prove  that  the  area  AD  EGA  is  equal  to  the  rectangle 
EHBG.  The  area  ADGBA  can  be  divided  in  two  parts,  the  tri- 
angle ABG  and  the  part  ADBCA  between  the  curve  and  the 
chord.  The  triangle  is  equal  to  the  rectangle  FIBG,  whileADBCA 
is  equal  to  two  thirds  of  the  parallelogram  MNBA,  and  hence 
equal  to  the  rectangle  EH  IF.  Both  together  are  therefore  equal 
to  the  rectangle  EHBG,  and  the  two  areas  between  the  stepping 
line  and  the  curve  on  both  sides  of  EH  are  thus  equal. 

If  the  curve  between  A  and  B  is  sup- 
posed to  be  a  parabola  with  its  axis  par- 
allel to  the  axis  of  ordinates  the  con- 
struction has  to  be  modified  a  little. 
Through  the  center  C  of  the  chord  AB 
(Fig.  76)  draw  a  vertical  line  CD  as  far 
as  the  parabola.  On  CD  find  the  point 
K  whose  distance  from  C  is  double  the 
distance  from  D  and  draw  through  it  a 
parallel  to  the  chord  AB.  This  parallel 

intersects  a  horizontal  line  through  C  at  a  point  L.  Then  EH 
must  pass  through  L.  This  may  be  shown  in  the  following  way. 
The  area  between  the  parabola  ADB  and  the  chord  AB  is  equal 
to  two  thirds  of  the  parallelogram  MNBA,  MN  being  the  tan- 
gent to  the  parabola  at  the  point  D.  If  D'  is  the  point  of  inter- 
section of  NN  and  the  horizontal  line  through  C,  we  have  evi- 
dently 

CL  =  f  CD'. 

Therefore  the  rectangle  EHIF  is  equal  to  the  area  ADB  A  be- 
tween the  parabola  and  the  chord  and  EHBG  is  equal  to  ADGBA. 
Any  part  of  a  curve  can  be  approximated  by  the  arc  of  a 
parabola  with  sufficient  accuracy  if  the  part  to  be  approximated 
is  sufficiently  small.  When  the  direction  of  the  curve  is  nowhere 
parallel  to  the  axis  of  coordinates,  both  kinds  of  parabolas  may 
be  used  for  approximation,  those  whose  axes  are  parallel  to  the 
axis  of  x  and  those  whose  axes  are  parallel  to  the  axis  of  y.  But 


FIG.  78. 


108 


GRAPHICAL  METHODS. 


when  the  direction  in  one  of  the  points  is  horizontal  (Fig.  76), 
we  can  only  use  those  with  vertical  axes  and  when  the  direction 
in  one  of  the  points  is  vertical  we  can  only  use  those  with  hori- 
zontal axes.  Accordingly  we  have  to  use  either  of  the  two  con- 
structions to  find  the  position  of  the  vertical  part  of  the  step. 
Do  not  draw  your  steps  too  small.  For,  although  the  difference 
between  the  broken  line  and  the  integral  curve  becomes  smaller, 
the  drawing  is  liable  to  an  accumulation  of  small  errors  owing 

to  the  considerable  number 
of  corners  of  the  broken 
line  and  little  errors  of 
drawing  committed  at  the 
corners.  Only  practical  ex- 
perience enables  one  to  find 
the  size  best  adapted  to 
the  method. 

Statical  moments  of  areas 
may  be  found  by  a  double 

graphical  integration.  Let  us  consider  the  area  between  the  curve 
y  =  /(*)  (Fig.  77),  the  axis  of  x  and  the  ordinates  corresponding 
to  x  =  0  and  x  =  £.  The  statical  moment  with  respect  to  the 
vertical  through  x  =  £  is  the  integral  of  the  products  of  each 
element  ydx  and  its  distance  £  —  x  from  the  vertical 


Fio.  77. 


Let  us  regard  M  as  a  function  of  £  and  differentiate  it: 


=  0  +  jf  ydx. 


That  is  to  say,  a  graphical  integration  of  the  curve  y  =  f(x~) 
beginning  at  x  =  0  furnishes  the  curve  whose  ordinate  is 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


109 


Hence  a  second  integration  of  this  latter  curve  will  furnish 
the  curve  M  as  a  function  of  £.  As  M  vanishes  for  £  =  0  the 
second  integration  must  also  begin  at  the  abscissa  x  =  0. 


, 

FIG.  78. 

Fig.  78  shows  an  example.  Each  ordinate  of  the  curve  found 
by  the  second  integration  is  the  statical  moment  of  the  area  on 
the  left  side  of  it  with  respect  to  the  vertical  through  this  same 
ordinate.  The  ordinate  furthest  to  the  right  is  the  statical 
moment  of  the  whole  area  with  respect  to  the  vertical  on  the 
right.  The  statical  moment  of  the  whole  area  with  respect  to  a 
vertical  line  through  any  point  x\  is  the  integral 


J^    (ari  -  x)ydx. 


Considered  as  a  function  of  xi  its  differential  coefficient  is 


t        J 

Kfe-. 


That  is  to  say,  the  differential  coefficient  is  independent  of  Xi, 
hence  the  statical  moment  is  represented  by  a  straight  line.  As 
its  differential  coefficient  is  represented  by  a  horizontal  line 
through  the  last  point  on  the  right  of  the  curve 


£>*• 


110  GRAPHICAL   METHODS. 

the  direction  of  the  straight  line  is  found  by  drawing  a  line 
through  P  and  through  the  point  of  intersection  Q  of  the  hori- 
zontal line  and  the  axis  of  ordinates  (Fig.  78).  The  position  of 
the  straight  line  is  then  determined  by  the  condition  that 


r 


(xi  -  x)ydx 
for  0-1=  |  is  equal  to  the  statical  moment 

£  -  x}ydx. 


We  have  therefore  only  to  draw  a  parallel  to  PQ  through  the 
last  point  R  of  the  curve  for  M(i-)  found  by  the  second  integration. 
The  ordinates  of  this  straight  line  for  any  abscissa  a'i  represent 
the  values  of 


measured  in  the  unit  of  length  of  the  ordinates.     The  point  of 
intersection  E  with  the  axis  of  x  determines  the  position  of  the 
vertical  in  regard  to  which  the  statical  moment  is  zero,  that  is  to 
say,  the  vertical  through  the  center  of  gravity. 
The  moment  of  inertia  of  the  area 


about  the  axis  x  =  £  is  found  in  a  similar  way.     It  is  exoressed 
by  the  integral 


Considered  as  a  function  of  £  we  find  by  differentiation 


-  x)ydx. 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS.  Ill 

That  is  to  say,  the  differential  coefficient  is  equal  to  double  the 
statical  moment  about  the  same  axis.  This  holds  for  every  value 
of  £.  Hence  we  obtain  %T  as  a  function  of  £  by  integrating 
the  curve  for  3f  (£).  For  £  =  0  we  have  T  =  0,  so  that  the  curve 
begins  on  the  axis  of  x  at  £  =  0, 
The  integral 

'"yd* 

is  zero  for  x  =  a.  The  curve  representing  the  integral  has  to 
intersect  the  axis  of  x  at  x  =  a  (admitting  values  of  x  >  a  and 
x  <  a),  and  it  is  there  that  we  begin  the  construction  of  the 
broken  line.  If  instead  we  begin  it  at  the  point  x  =  a,  y  =  c, 
the  only  difference  is  that  the  whole  integral  curve  is  shifted 
parallel  to  the  axis  of  ordinates  by  an  amount  equal  to  c  upwards 
if  c  is  positive,  downwards  if  it  is  negative.  But  the  form  of  the 
curve  remains  the  same.  It  is  different  when  this  curve  is 
integrated  a  second  time.  For  instead  of 


f' 


* ydx 

we  now  integrate 

ydx  +  c. 


r« 


The  ordinate  of  the  integral  curve  is  therefore  changed  by  an 
amount  equal  to  c(x  —  a)  and  besides  if  the  second  integral  curve 
is  begun  at  x  =  a,  y  =  c\  instead  of  x  =  a,  y  =  0  the  change 
amounts  to 

c(x  —  a)  +  ci, 

so  that  the  difference  between  the  ordinates  of  the  new  integral 
curve  and  the  ordinates  of  the  straight  line 

y  =  c(x  —  a)  +  ci 

is  equal  to  the  ordinates  of  the  first  integral  curve  (Fig.  79). 

This  effect  of  adding  a  linear  function  to  the  ordinates  of  the 
integral  curve  is  also  attained  by  shifting  the  pole  P  upward  or 


112 


GRAPHICAL   METHODS. 


downward.  For  it  evidently  comes  to  the  same  thing  whether 
the  curve  to  be  integrated  is  shifted  upward  by  the  amount  c  or 
whether  the  point  P  is  moved  downward  by  the  same  amount,  so 
that  the  relative  position  of  P  and  the  curve  to  be  integrated 
is  the  same  as  before.  Changing  the  ordinate  of  P  by  —  c  adds 


'  fff(x)dxdx 
'    "" 


FIG.  79. 

c(x  —  a)  to  the  ordinates  of  the  integral  curve.  c(x  —  a)  is  the 
ordinate  of  a  straight  line  parallel  to  the  straight  line  from  the 
new  position  of  P  to  the  origin. 

By  this  device  of  shifting  the  position  of  P  upward  or  down- 
ward the  integral  curve  may  sometimes  be  kept  within  the 
boundaries  of  the  drawing  without  any  reduction  of  the  scale  of 
ordinates.  A  good  rule  is  to  choose  the  ordinate  of  P  about 
equal  to  the  mean  ordinate  of  the  curve  to  be  integrated.  The 
ordinates  of  the  integral  curve  will  then  be  nearly  the  same  at 
both  ends.  The  value  of  the  integral 


f 


ydx 


is  equal  to  the  difference  between  the  ordinates  of  the  integral 
curve  and  the  ordinates  of  a  straight  line  parallel  to  PO  through 
the  point  of  the  integral  curve  whose  abscissa  is  a. 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS.  113 

When  the  ordinate  of  P  is  accurately  equal  to  the  mean  ordinate 
of  the  curve  to  be  integrated  for  the  interval  x  =  a  to  b  the 
ordinates  of  the  integral  curve  will  be  accurately  the  same  at  the 
two  ends.  But  we  do  not  know  the  mean  ordinate  before  having 
integrated  the  curve. 

After  having  integrated  we  find  the  mean  ordinate  for  the 
interval  x  =  a  to  6  by  drawing  a  straight  line  through  P  parallel 
to  the  chord  AB  of  the  integral  curve,  A  and  B  belonging  to  the 
abscissas  x=a  and  x=b.     This 
line  intersects  the  axis  of  ordi- 
nates at  a  point  whose  ordinate 
is  the  mean  ordinate. 

Suppose  a  beam  AB  is  sup- 
ported at  both  ends  and  loaded 
by  a  load  distributed  over  the 

beam  as  indicated  by  Fig.  80.  That  is  to  say,  the  load  on  dx  is 
measured  by  the  area  ydx.  Let  us  integrate  this  curve  graph- 
ically, beginning  at  the  point  A  with  P  on  the  line  AB.  The 
final  ordinate  at  B 


f. 


ydx 


gives  the  whole  load  and  is  therefore  equal  to  the  sum  of  the  two 
reactions  at  A  and  B  that  equilibrate  the  load.  Integrating  this 
curve  again  we  obtain  the  curve  whose  ordinate  is  equal  to 


Y  being  written  for 

(* 

ydx 


J 


The  ordinate  of  this  curve  at  any  point  x  =  £  represents  the 
statical  moment  of  the  load  between  the  verticals  x  =  a  and 
x  =  £  about  the  axis  x  =  £.  Its  final  ordinate  BM,  Fig.  81,  is 
the  moment  of  the  whole  load  about  the  point  B,  and  as  the  reac- 
tions equilibrate  the  load  it  must  be  equal  to  the  moment  of  the 


114 


GRAPHICAL   METHODS. 


reactions  about  the  same  point  and  therefore  opposite  to  the 
moment  of  the  reaction  at  A  about  B.  If  the  reaction  at  A  is 
denoted  by  Fa  we  therefore  have 

Fa(b  -  a)  =    C  Ydx. 

That  is  to  say,  Fa  is  equal  to  the  mean  ordinate  of  the  curve 
F  = 

in  the  interval  x  =  a  to  6.  The  mean  ordinate  is  found  by 
drawing  a  parallel  to  AM  through  P  which  intersects  the  vertical 
through  A  at  the  point  F  so  that  AF  =  Fa.  As  DB  is  equal  to 


FIG.  81. 


the  sum  of  the  two  reactions  a  horizontal  line  through  F  will 
divide  BD  into  the  two  parts  BG  =  Fa  and  GD  =  F*. 

Shifting  the  position  of  P  to  P'  on  the  horizontal  line  FG 
and  repeating  the  integration 


f 

Jo, 


Ydx, 


we  obtain  a  curve  with  equal  ordinates  at  both  ends.  If  we 
begin  at  A  it  must  end  in  B.  Its  ordinates  are  equal  to  the 
difference  between  the  ordinates  of  the  chord  AM  and  the  curve 
AM  (Fig.  81),  and  represent  the  moment  about  any  point  of 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


115 


the  beam  of  all  the  forces  on  one  side  of  the  point  (load  and 
reaction). 

The  area  of  a  closed  curve  may  be  found  by  integrating  over  the 
whole  boundary.  Suppose  x  =  a  and  x  =  b  t6  be  the  limits  of 
the  abscissas  of  the  closed  curve,  the  vertical  x  =  a  touching  the 
curve  at  A  and  the  vertical  x  =  b  at  B  (Fig.  82).  By  A  and  B 
the  closed  curve  is  cut  in  two,  both  parts  connecting  A  and  B. 
Let  us  denote  the  upper  part  by  y  =  f\(x)  and  the  lower  part 
by  y  =  fz(x).  The  whole  area  is  then  equal  to  the  difference 


FIG.  82. 


jT  ftWdx  -  £  f*(x)dx, 


or  equal  to 


We  begin  the  integral  curve. over  the  upper  part  at  the  vertical 
x  =  a  at  a  point  E,  the  ordinate  of  which  is  arbitrary,  and  draw 
the  broken  line  as  far  as  F  on  the  vertical  x  =  b  (Fig.  82).  Then 
we  integrate  back  again  over  the  lower  part,  continuing  the 
broken  line  from  F  to  G.  The  line  EG  measured  in  the  unit  of 
length  set  down  for  the  ordinates  is  equal  to  the  area  measured 
in  units  of  area,  this  unit  being  a  rectangle  formed  by  PO  and 
the  unit  of  ordinates.  That  is  to  say,  the  area  is  equal  to  the 
area  of  a  rectangle  whose  sides  are  PO  and  EG. 


116 


GRAPHICAL   METHODS. 


The  method  is  not  limited  to  the  case  drawn  in  Fig.  82,  where 
the  closed  curve  intersects  any  vertical  not  more  than  twice.  A 
more  complicated  case  is  shown  in  Fig.  83.  But  in  all  those  cases 


FIQ.  83. 


where  the  object  is  not  to  find  the  integral  curve  but  only  to  find 
the  value  of  the  last  ordinate  the  method,  cannot  claim  to  be 
of  much  use,  because  it  cannot  compete  with  the  planimeter. 


FIQ.  84. 


For  the  construction  of  the  broken  line  we  have  drawn  the 
steps  in  such  a  manner  that  the  areas  on  both  sides  of  the  vertical 
part  of  a  step  between  the  curve  and  the  stepping  line  are  equal. 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS. 


117 


It  would  have  also  been  admissible  to  construct  the  stepping 
line  in  such  a  way  that  the  areas  on  both  sides  of  the  horizontal 
part  of  a  step  are  equal  (Fig.  84).  Only  the  broken  line  would 
consist  of  a  series  of  chords  instead  of  a  series  of  tangents  of  the 
integral  curve.  The  points  Ka,  Kb,  •  •  •,  where  the  horizontal 
parts  of  the  steps  intersect  the  curve  would  determine  the  ab- 
scissas of  the  points  of  the  integral  curve,  where  its  direction  is 
parallel  to  the  direction  of  the  broken  line.  But  this  forms  very 
little  help  for  drawing  the  integral  curve.  That  is  the  reason 
why  the  former  method  where  the  broken  line  consists  of  a  series 
of  tangents  is  to  be  preferred.  However  where  the  object  is  only 
to  find  the  last  ordinate  of  the  integral  curve  the  two  methods 
are  equivalent. 

§  14.  Graphical  Differentiation. — The  graphical  differentiation 
of  a  function  represented  by  a  curve  is  not  so  satisfactory  as  the 
graphical  integration  because 
the  values  of  the  differential 
coefficient  are  generally  not 
very  well  defined  by  the  curve. 
The  operation  consists  in 
drawing  tangents  to  the  given 
curve  and  drawing  parallels 
through  P  to  the  tangents 
(Fig.  85).  The  points  of  in- 
tersection of  these  parallels 
with  the  axis  of  ordinates  fur- 
nish the  ordinates  of  the  curve  representing  the  derivative. 
The  abscissa  to  each  ordinate  coincides  with  the  abscissa  of  the 
point  of  contact  of  the  corresponding  tangent.  The  principal 
difficulty  is  to  draw  the  tangent  correctly.  As  a  rule  it  can  be 
recommended  to  draw  a  tangent  of  a  given  direction  and  then 
mark  its  point  of  contact  instead  of  trying  to  draw  the  tangent 
for  a  given  point  of  contact.  A  method  of  finding  the  point  of 
contact  more  accurately  than  by  mere  inspection  consists  in 
drawing  a  number  of  chords  parallel  to  the  tangent  and  to 


FIG.  85. 


118 


GRAPHICAL    METHODS. 


bisect  them.  The  points  of  bisection  form  a  curve  that  inter- 
sects the  given  curve  at  the  point  of  contact  (Fig.  86).  When  a 
number  of  tangents  are  drawn,  their  points  of  contact  marked 
and  the  points  representing  the  differential  coefficient  constructed, 

the  derivative  curve  has  to  be 
drawn  through  these  points. 
This  may  be  done  more  accur- 
ately by  means  of  the  stepping 
line.  The  horizontal  parts  of 
the  steps  pass  through  the 
points  while  the  vertical  parts 
lie  in  the  same  vertical  as  the 
point  of  intersection  of  two 

consecutive  tangents.  The  derivative  curve  connects  the  points 
in  such  a  way  that  the  areas  between  it  and  the  stepping  line  are 
equal  on  both  sides  of  the  vertical  parts  of  each  step.  Thus 
the  result  of  the  graphical  differentiation  is  exactly  the  same 


Fia.  87. 


figure  that  we  get  by  integration,  only  the  operations  are  carried 
out  in  the  inverse  order. 

A  change  of  the  distance  PO  (Fig.  87)  changes  the  ordinates 
of  the  derivative  curve  in  the  same  proportion  and  for  the  same 
reason  that  it  changes  the  ordinates  of  the  integral  curve  when  we 


DIFFERENTIAL   AND    INTEGRAL    CALCULUS.  119 

are  integrating,  but  in  the  inverse  ratio.  Any  change  of  the  or- 
dinate  of  P  only  shifts  the  curve  up  or  down  by  an  equal  amount, 
so  that  if  we  at  the  same  time  change  the  axis  of  x  and  draw  it 
through  the  new  position  of  P  the  ordinates  of  the  curve  will 
remain  the  same  and  will  represent  the  differential  coefficient. 

When  a  function  f(x,  y)  of  two  variables  is  given  by  a  diagram 
showing  the  curves  f(x,  y)  =  const,  for  equidistant  values  of 
f(x,  y)  the  partial  differential  coefficients  can  be  found  at  any 
point  XQ,  2/0  by  means  of  drawing  curves  whose  ordinates  represent 
f(x,  yo)  to  the  abscissa  x  orf(xQ,  y)  to  the  abscissa  y  and  applying 
the  methods  explained  above.  For  this  purpose  a  parallel  is 
drawn  to  the  axis  of  x,  for  instance,  through  the  point  XQ,  y0 
and  at  the  points  where  it  intersects  the  curves /(or,  y)  =  const, 
ordinates  are  erected  representing  the  values  of  f(x,  y0)  in  any 
convenient  scale.  A  smooth  curve  is  then  drawn  though  the 
points  so  found  and  the  tangent  of  the  curve  at  the  point  XQ 
furnishes  the  differential  coefficient  df/dx  for  x  =  XQ,  y  =  y0. 

The  differential  coefficients  df/dx,  df/dy  are  best  represented 
graphically  by  a  straight  line  starting  from  the  point  x,  y  to 
which  the  differential  coefficients  correspond,  and  of  such  length 
and  direction  that  its  orthogonal  projections  on  the  axis  of  x 
and  y  are  equal  to  df/dx  and  df/dy.  This  line  represents  the 
gradient  of  the  function  f(x,  y)  at  the  point  x,  y.1  It  is  normal 
to  the  curve  f(x,  y)  =  const,  that  passes  through  the  point  x,  y, 
its  direction  being  the  direction  of  steepest  ascent.  Its  length 
measures  the  slope  of  the  surface  z  =  f(x,  y)  in  the  direction  of 
steepest  ascent.  This  is  shown  by  considering  the  slope  in  any 
other  direction.  Let  us  change  x  and  y  by 

r  cos  a,     r  sin  a 
and  consider  the  corresponding  change 

Az  =  f(x  +  r  cos  a,y+r  sin  a)  —  f(x,  y) 
of  the  function.     By  Taylor's  theorem  we  can  write  it 
1  See  Chap.  II,  §  10. 


120  GRAPHICAL   METHODS. 

—  r  cos  a  +  —  r  sin  a.  +  terms  of  higher  order  in  r, 
ox  oy 

a  is  the  direction  from  the  point  x,  y  to  the  new  point  x  -\-  r  cos  a, 
t/  +  r  sin  a  and  r  is  the  distance  of  the  two  points.  Dividing 
Az  by  r  and  letting  r  approach  to  zero  we  find 

AS     a/          .  a/  . 

hm  —  =  —  cos  a  +  —  sin  a. 
r       dx  By 

This  expression  measures  the  slope  of  the  surface  z  =  f(xy] 
in  the  direction  a.  Now  let  us  introduce  the  length  /  and  the 
angle  X  of  the  gradient,  and  write 

~  =  I  cos  X.     ~  =  I  sin  X. 
dx  dy 

Then  we  have 

—  cos  a  +  —  sin  a  =  I  cos  (a  —  X). 
dx  dy 

That  is  to  say,  the  slope  in  any  direction  a  is  proportional  to 
cos  (a  —  X),  it  is  a  maximum  in  the  direction  of  the  gradient 
(a  =  X)  and  zero  in  a  direction  perpendicular  to  it  and  negative 
in  all  directions  that  form  an  obtuse  angle  with  it.  When  all 
three  coordinates  are  measured  in  the  same  unit,  the  length  of 
I  measured  in  this  unft  is  equal  to  the  tangent  of  the  angle  of 
steepest  ascent.  Hence  the  length  of  the  gradient  varies  with 
the  unit  of  length.  When  the  unit  of  length  in  which  the  values 
of  f(xy)  are  plotted  is  kept  unaltered,  while  we  change  the  unit 
of  length  corresponding  to  the  values  x  and  y,  the  length  of  the 
gradient  varies  with  the  square  of  the  unit  of  length. 

§  15.  Differential  Equations  of  the  First  Order.  —  In  the  problem 
of  solving  a  differential  equation  of  the  first  order 


by  graphical  methods  the  first  question  is  how  to  represent 
the  differential  equation  graphically.  If  x  and  y  are  meant  to 
be  the  values  of  rectangular  coordinates,  the  geometrical  meaning 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS. 


121 


of  the  differential  equation  is  that  at  every  point  x,  y,  where 
f(x,  y}  is  defined,  the  equation  prescribes  a  certain  direction  for 
the  curve  that  satisfies  it.  Let  us  suppose  curves  drawn  through 
all  those  points  for  which  f(x,  y)  has  certain  constant  values. 
Each  curve  then  corresponds  to  a  certain  direction  or  the  opposite 
direction.  Let  us  distinguish  the  curves  by  different  numbers  or 
letters  and  let  us  draw  a  pencil  of  rays  together  with  the  curves 
and  mark  the  rays  with  the  same  numbers  or  letters  in  such  a  way 
that  each  of  them  shows  the  direction  corresponding  to  the 


FIG.  88. 

curve  marked  with  that  particular  number  or  letter  (Fig.  88). 
Our  drawing  of  course  only  comprises  a  certain  region  in  which 
we  propose  to  find  the  curves  satisfying  the  differential  equation. 
It  may  be  that  f(xy)  is  defined  beyond  the  boundaries  of  our 
drawing.  Those  regions  have  to  be  dealt  with  separately. 

The  graphical  representation  of  the  differential  equation  in 
the  region  considered  consists  in  the  correspondence  between 
the  curves  and  the  rays.  It  is  important  to  observe  that  this 
representation  is  independent  of  the  system  of  coordinates  by 
means  of  which  we  have  deduced  the  curves  from  the  equation 


122  GRAPHICAL   METHODS. 

We  can  now  introduce  any  system  of  coordinates  £,  77  and  find 
from  our  drawing  the  equation 


that  is  to  say,  we  can  find  the  value  of  <p(%,  77)  at  any  point  £,  77 
of  our  drawing.  If,  for  instance,  the  unit  of  length  is  the  same 
for  £  and  77  we  draw  a  line  through  the  center  of  the  pencil  of  rays 
in  the  direction  of  the  positive  axis  of  £  and  a  line  perpendicular  to 
it  at  the  distance  1  from  the  center.  The  segment  on  the  second 
line  between  the  first  line  and  the  point  of  intersection  with  one 
of  the  rays  measured  in  units  of  length  and  counted  positive  in 
the  direction  of  positive  77  furnishes  the  value  of  ?>(£,  77)  for  all 
the  points  £,  77  corresponding  to  that  particular  ray.  In  this 
respect  the  graphical  representation  of  a  differential  equation 
is  superior  to  the  analytical  form,  in  which  certain  coordinates 
are  used  and  the  transformation  to  another  system  of  coordinates 
requires  a  certain  amount  of  calculation. 

Now  let  us  try  to  find  the  curve  through  a  given  point  P  on 
the  curve  marked  (a)  (Fig.  88)  that  satisfies  the  differential  equa- 
tion. We  begin  by  drawing  a  series  of  tangents  of  a  curve 
that  is  meant  to  be  a  first  approximation.  Through  P  we  draw 
a  parallel  to  the  ray  (a)  as  far  as  the  point  Q  somewhere  in  the 
middle  between  the  curves  (a)  and  (6).  Through  Q  we  draw  a 
parallel  to  the  ray  (6)  as  far  as  R  somewhere  in  the  middle 
between  the  curves  (&)  and  (c).  Through  R  we  again  draw  a 
parallel  to  the  ray  (c)  and  so  on.  The  curve  touching  this 
broken  line  at  the  points  of  intersection  with  the  curves  (a), 
(6),  •  •  •  is  a  first  approximation.  But  we  need  not  draw  this 
curve.  In  order  to  find  a  better  approximation  we  introduce  a 
rectangular  system  of  coordinates  x,  y,  laying  the  axis  of  x  some- 
what in  the  mean  direction  of  the  broken  line.  Let  us  denote 
by  i/i  the  function  of  x  that  corresponds  to  the  curve  forming  the 
first  approximation.  The  second  approximation  yz  is  then  ob- 
tained as  an  integral  curve  of  f(x,  y\},  that  is,  of  dyi/dx 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS.         123 


I 

Jx 


denoting  by  xp,  yp,  the  coordinates  of  P.  For  this  purpose  the 
curve  whose  ordi  nates  are  equal  to  f(x,  yi)  or  dyi/dx  has  to  be  con- 
structed first.  The  values  of  f(x,  yi)  are  found  immediately  at 
the  points  where  the  first  approximation  intersects  the  curve 
(a),  (6)  •  •  •  by  differentiation  in  the  way  described  above.  A 
line  is  drawn  through  the  center  of  the  pencil  of  rays  parallel  to 
the  axis  of  x  and  a  line  perpendicular  to  it  at  a  convenient  dis- 
tance from  the  center.  This  distance  is  chosen  as  the  unit  of 
length.  The  points  of  intersection  of  this  line  with  the  rays  de- 
termine segments  whose  lengths  are  equal  to  the  values  of  f(x,  yi) 
on  the  corresponding  curves.  These  values  are  plotted  as  ordi- 
nates  to  the  abscissas  of  the  points  where  the  first  approximation 
intersects  the  curves  (a),  (6),  •  •  •  and  a  curve 


is  drawn  (Fig.  88).  This  curve  is  integrated  graphically  begin- 
ning at  the  point  P  and  the  integral  curve  is  a  second  approxi- 
mation. Again  we  need  not  draw  the  curve.  The  broken  line 
suffices,  if  we  intend  to  construct  a  third  approximation.  In 
this  case  we  have  to  repeat  the  foregoing  operation.  This  can 
now  be  performed  much  quicker  than  in  the  first  case  because  the 
values  of  f(x,  y)  on  the  curves  (a),  (6),  •  •  •  have  already  been 
constructed  and  are  at  our  disposal.  In  order  to  find  the  curve 


we  have  only  to  shift  the  same  ordinates  to  new  abscissas  and 
make  these  coincide  with  the  abscissas  of  the  points  where  the 
second  approximation  intersects  the  curves  (a),  (6),  •••.  The 
curve 


is  then  drawn  and  integrated  graphically,  beginning  at  the  point 
P. 


124  GRAPHICAL   METHODS. 

Suppose  now  the  integral  curve  did  not  differ  from  the  second 
approximation,  it  would  mean  that 

2/2  =  yP  +  J    f(x,y2)dx, 
or  that 


that  is  to  say,  that  yz  satisfies  the  differential  equation. 

If  there  is  a  perceptible  difference  the  integral  curve  represents 
a  third  approximation.  It  has  been  shown  by  Picard  that  pro- 
ceeding in  this  way  we  find  the  approximations  (under  a  certain 
condition  to  be  discussed  presently)  converging  to  the  true  solu- 
tion of  the  differential  equation,  so  that  after  a  certain  number 
of  operations  the  error  of  the  approximation  must  become 
imperceptible. 

Denoting  by  yn  the  function  of  the  nth  approximation  we  have 

2/n+i  =  yP  +  I  f(x,  yn)dx. 

y. 

The  true  solution  writh  the  same  initial  conditions  y  =  yp  for 
x  =  Xp  satisfies  the  equation 

cx 

y  =  yp  +  I  /fo  y}™- 

Hence 

yn+1  -  y  =  J    [f(x,  yn)  -  f(x,  y)]dx, 
or 


Let  us  now  suppose  that  the  absolute  value  of 

f(x,  yn)  —  f(x,  y) 

yn-y 

for  all  the  values  of  x,  y,  yn  within  the  considered  region  does 


DIFFERENTIAL  AND   INTEGRAL  CALCULUS.  125 

not  surpass  a  certain  limit  M,  then  it  follows  that  a  certain  relation 
must  exist  between  the  maximum  error  of  yn,  which  we  denote  by 
en  and  the  maximum  error  of  yn+i,  which  we  denote  by  en+i* 
The  absolute  value  of  the  integral  not  being  larger  than 


(  |  x  —  xn  |  denoting  the  absolute  value  of  x  —  xn}  we  have 
en+i  ^  M  |  x  —  xn  |  en. 

Hence  as  long  as  the  distance  x  —  xn  over  which  the  integration 
is  performed  is  so  small  that 

M  |  X  -  Xn  |    ^   k  <   1, 

k  being  a  constant  smaller  than  one,  the  error  of  yn+i  cannot  be 
larger  than  a  certain  fraction  of  the  maximum  error  of  yn. 
But  in  the  same  way  it  follows  that  the  error  of  yn  cannot  be 
larger  than  the  same  fraction  of  the  maximum  error  of  yn-i,  and 

so  on,  so  that 

en+i  £  ken  £  kzen-i  •  •  •  £  k»ei. 

But  as  ei  is  a  constant  and  k  a  constant  smaller  than  one,  knei 
must  be  as  small  as  we  please  for  a  sufficient  large  value  of  n. 
That  is  to  say,  the  approximations  converge  to  the  true  solution. 
M  being  a  given  constant  the  condition  of  convergence 

M  |  x  -  xp  |  <  k  <  1 

limits  the  extent  of  our  integration  in  the  direction  of  the  axis  of  x. 
But  it  does  not  limit  our  progress.  From  any  point  Pr  that  we 
have  reached  with  sufficient  accuracy  we  can  make  a  fresh  start, 
choosing  a  new  axis  of  x  suited  to  the  new  situation.  As  a 
rule  it  does  not  pay  to  trouble  about  the  value  of  M  and  to  try 
to  find  the  extent  of  the  convergence  by  the  help  of  this  value. 
The  actual  construction  of  the  approximations  will  show  clearly 
enough  how  far  to  extend  the  integration.  As  far  as  two  consecu- 
tive approximations  show  no  difference  they  represent  the  true 
curve. 


126  GRAPHICAL   METHODS. 

Suppose  that 


has  the  same  sign  for  all  values  x,  y,  yn  concerned.  Say  it  is 
negative.  Suppose  further  that  yn  —  y  is  of  the  same  sign  for 
the  whole  extent  of  the  integration 


that  is  to  say,  the  approximative  curve  yn  is  all  on  one  side  of  the 
true  curve.  Then  if  x  —  xp  is  positive,  ynn  ~  y  must  evidently 
be  of  the  opposite  sign  from  yn  —  y,  or  the  approximative  curve 
yn+i  is  all  on  the  other  side  of  the  true  curve  from  yn.  For  these 
and  all  following  approximations  the  true  curve  must  lie  between 
two  consecutive  approximations.  If  the  first  approximation  y±  is 
all  on  one  side  of  the  true  curve  the  theorem  holds  for  any  two 
consecutive  approximations.  This  is  very  convenient  for  the  esti- 
mation of  the  error. 
In  Fig.  88 


y»  -  y 

is  negative  from  the  point  P  as  far  as  somewhere  near  S.  The 
first  approximation  is  all  on  the  upper  side  of  the  true  curve. 
Therefore  the  second  approximation  must  be  below  the  true 
curve  at  least  as  far  as  somewhere  near  S. 

When  the  sign  is  positive  the  same  theorem  holds  for  negative 
values  of  x  —  xp.  If  the  integration  has  been  performed  in  the 
positive  direction  of  x,  it  may  be  a  good  plan  to  check  the  result 
by  integrating  backwards,  starting  from  a  point  that  has  been 
reached  and  to  try  if  the  curve  gets  back  to  the  first  starting 
point.  In  this  direction  we  profit  from  the  advantage  of  the 
true  curve  lying  between  consecutive  approximations  and  are 
better  able  to  estimate  the  accuracy  of  our  drawing. 

We  have  seen  that  the  convergence  depends  on  the  maximum 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS.  127 

absolute  value  of 

f(x,  yn)  —  f(x,  y) 

yn-  y 

for  all  values  of  x,  y,  yn  concerned.     In  order  to  find  the  maximum 
value  we  may  as  well  consider 


for  all  values  of  x,  y  within  the  region  considered.  For  if  we 
assume  df/dy  to  be  a  continuous  function  of  y,  it  follows  that 
the  quotient  of  differences 


yn-  y 

must  be  equal  to  df/dy  taken  for  the  same  value  of  x  and  a  value 
of  y  between  y  and  yn.  This  is  immediately  seen  by  plotting 
f(x,  y)  as  ordinate  to  the  abscissa  y  for  a  fixed  value  of  x.  The 
value  of  the  quotient  of  differences  is  determined  by  the  slope 
of  the  chord  between  the  two  points  of  abscissas  y  and  yn.  The 
slope  of  the  chord  is  equal  to  the  slope  of  the  curve  at  a  certain 
point  between  the  ends  of  the  chord.  The  value  of  df/dy  at  this 
point  is  equal  to  the  value  of 


yn  —  y 

Now  let  us  consider  how  the  coordinate  system  may  be  chosen 
in  order  to  make  df/dy  as  small  as  possible  and  thus  obtain  the 
best  convergence.  For  this  purpose  let  us  investigate  how  the 
value  of  df/dy  changes  at  a  certain  point,  when  the  system  of 
coordinates  is  changed. 

Let  us  start  with  a  given  system  of  rectangular  coordinates  £, 
77  with  which  the  differential  equation  is  written 


The  direction  of  the  curve  satisfying  the  differential  equation 


128  GRAPHICAL   METHODS. 

forms  a  certain  angle  a  with  the  positive  axis  of  £  determined  by 

tg  a  =  f$  =  ^'  ^ 

(assuming  the  coordinates  to  be  measured  in  the  same  unit). 
Now  let  us  introduce  a  new  system  of  rectangular  coordinates 
x,  y  connected  with  the  system  £,  rj  by  the  equations 

x  =  %  cos  co  +  TJ  sin  co, 
y  =  —  £  sin  co  -f-  T/  cos  co, 
which  are  equivalent  to 

£  =  x  cos  co  —  y  sin  co, 
77  =  x  sin  co  +  y  cos  cc, 

co  being  the  angle  between  the  positive  direction  of  x  and  the 
positive  direction  of  £,  counted  from  £  towards  x  in  the  usual  way. 
The  angle  formed  by  the  direction  of  the  curve  with  the  positive 
direction  of  the  axis  of  x  is  a  —  co,  and  therefore 

g=tg  (a -co)  =/(*,</). 

Consequently  we  obtain  for  a  given  value  of  w 

df  =  1  da 

dy  ~  cos2  (a  —  co)  dy ' 

or  remembering  that  a  is  given  as  a  function  of  £  and  17, 

df  1  (       da    .          ,    da  \ 

—  =  — IT, N '  I  —  —  sin  co  -f-  —  cos  co  I. 

dy       cos2  (a  —  co)   V       d£  &l  ) 

For  simplicity's  sake  we  shall  assume  that  the  axis  of  £  is  the 
tangent  of  the  curve  <p(£,i\)  =  const,  that  passes  through  the 
given  point,  so  that  dafd!-  =  0. 
We  then  have 

df  1  da 

—  =  — r7 :  T-  cos  co, 

dy       cosj  (a  —  co)  drj 

and  our  object  is  to  find  how  df/dy  varies  for  different  values  of 


DIFFERENTIAL   AND    INTEGRAL    CALCULUS. 


129 


co.  The  value  of  da/dy  is  independent  of  co;  it  denotes  the  value 
of  the  gradient  of  a,  which  we  represent  by  a  straight  line  drawn 
from  the  origin  A  (Fig.  89)  perpendicular  to  the  curve  a  =  const, 
or  <f>(£,  17)  =  const. 

It  is  no  restriction  to  assume  the  value  of  da/drj  positive;  it 
only  means  that  the  direction  of  the  positive  axis  of  t]  is  chosen 


FIG.  89. 

in  the  direction  of  the  gradient.  Let  us  draw  the  line  AB  (Fig. 
89)  in  the  direction  of  the  positive  axis  of  £  and  of  the  same  length 
as  the  gradient. 

In  order  to  show  the  values  of  df/dy  for  the  different  positions 
of  the  axis  of  x  let  us  lay  off  the  value  of  dffdy  as  an  abscissa. 
For  instance  for  co  =  or,  df/dy  assumes  the  value 


da 

—  cos  a. 
dr, 

The  abscissa  corresponding  to  this  value  is  AB'  (Fig. 
10 


),  the 


130 


GRAPHICAL  METHODS. 


orthogonal  projection  of  AB  on  the  axis  of  x.  For  any  other 
position  AC  (Fig.  89)  corresponding  to  some  other  value  of  to, 
we  find  da/dri  cos  to  by  orthogonal  projection  of  AB  on  AC.  Then 
the  division  by  cos  (a  —  to)  furnishes  AC'  and  a  second  division 
by  cos  (a  —  to)  leads  to  AC.  Thus  a  certain  curve  can  be 
constructed  whose  polar  coordinates  are  r  =  df/dy  and  to,  the 
equation  in  polar  coordinates  being 

da  cos  to  ..„      da 

r  —  —  • 5-7 r-     or     [r  cos  (or  —  to)lz=  —  r  cos  to. 

drj      COS2  (a  —  to)  drj 

In  rectangular  coordinates  £,  y  the  equation  assumes  the  form 
(cos  a£  +  sin  arj)2  =  —  £. 

This  shows  that  the  equation  is  a  parabola,  the  axis  of  which  is 
perpendicular  to  the  direction  a.  AB'  is  a  chord  and  the  gradient 


FIG.  90. 

AG  is  a  tangent  of  the  parabola.  Bisecting  AB'  in  E,  drawing  EK 
perpendicular  to  AB'  as  far  as  the  axis  of  77  and  bisecting  EK  in 
D,  we  find  D  the  apex  of  the  parabola.  The  three  points  A,B',D 
together  with  the  gradient  will  suffice  to  give  us  an  idea  of  the 
size  and  sign  of  df/dy  for  the  different  positions  of  the  positive 
axis  of  x. 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS.  131 

df/dy  vanishes  when  the  axis  of  x  is  perpendicular  to  the  curve 
a  =  const.,  so  that  it  seems  as  if  this  were  the  most  favorable 
position.  We  must,  however,  bear  in  mind  that  the  axis  of  x 
is  kept  unaltered  for  a  certain  interval  of  integration.  When  we 
pass  on  to  other  points  the  axis  of  x  is  no  longer  perpendicular 
to  the  curve  a  =  const,  there.  The  position  of  the  axis  of  x  is 
good  when  the  average  value  of  df/dy  is  small.  In  Fig.  90  the 
parabolas  are  constructed  for  a  number  of  points  on  the  first 
approximation  of  a  curve  satisfying  the  differential  equation. 

If  we  want  to  make  use  of  the  parabolas  to  give  us  the  numerical 
values  of  df/dy  the  unit  of  length  must  also  be  marked  in  which 
the  coordinates  are  measured.  The  numerical  value  of  df/dy 
varies  as  the  unit  of  length  and  therefore  the  length  of  the  line 
representing  it  must  vary  as  the  square  of  the  unit  of  length. 
But  if  we  draw  a  line  whose  length  measured  in  the  same  unit  is 

equal  to  -TTTT  >  this  line  would  be  independent  of  the  unit  of 
length.  For  if  I  is  the  line  representing  the  unit  of  length  and 
I',  I"  the  lines  representing  the  values  df/dy  and 


would  be  the  ratio  l'/l  and  T-  the  ratio  I"  /I;   hence  I"  =  P/l'. 


Since  I'  varies  as  P  with  the  change  of  the  unit  of  length  I"  is 
independent  of  the  unit  of  length.  This  line  I"  represents  the 
limit  beyond  which  the  product 


becomes  greater  than  1.     If  df/dy  remained  the  same  this  would 
mean  the  limit  beyond  which  the  convergence  of  the  process  of 

approximation  ceases.      We  might  lay  off  the  length  of  TTT-  in 


the  different  directions  in  the  same  way  as  df/dy  has  been  laid 
off.  The  result  is  a  curve  corresponding,  point  by  point,  to  the 
parabola,  the  image  of  the  parabola  according  to  the  relation  of 
reciprocal  radii.  But  all  these  preparations  .as  a  rule  would  not 


132  GRAPHICAL  METHODS. 

pay.  It  is  better  to  attack  the  integration  at  once  with  an  axis 
of  x  somewhat  perpendicular  to  the  curves  a  =  const,  as  long 
as  the  direction  of  the  curve  forms  a  considerable  angle  with 
the  curve  a  =  const,  and  to  lose  no  time  in  troubling  about  the 
very  best  position.  The  convergence  wrill  show  itself,  wThen  the 
operations  are  carried  out.  When  the  angle  between  the  direction 
of  the  curve  that  satisfies  the  differential  equation  and  the  curve 
a  =  const,  becomes  small  the  apex  of  the  parabola  moves  far 
away  and  when  the  direction  coincides  with  that  of  the  curve 
a  =  const,  the  parabola  degenerates  into  two  parallel  lines  per- 
pendicular to  the  direction  of  the  curve  a.  =  const.  In  this  case 
the  best  position  for  the  axis  of  x  is  in  the  direction  of  the  curve 
a  —  const.  Without  going  into  any  detailed  investigation  about 
the  best  position  of  the  axis  of  x  we  can  establish  the  general  rule 
not  to  make  the  axis  of  x  perpendicular  to  the  direction  of  the 
curve  satisfying  the  differential  equation,  that  is  to  say,  not  to 
make  it  parallel  to  the  axis  of  the  parabola.  But  we  hardly  need 
pronounce  this  rule.  In  practice  it  would  enforce  its  own  observ- 
ance, because  for  that  position  of  the  axis  of  x  not  only  df/dy  but 
also  f(x,  y)  are  infinite  and  it  would  become  impossible  to  plot 
the  curve  Y  =  f(x,  y\). 

There  is  another  graphical  method  of  integrating  a  differential 
equation  of  the  first  order 


which  in  some  cases  may  well  compete  with  the  first  method. 
Like  the  first  it  is  the  analogue  of  a  certain  numerical  method. 
The  numerical  method  starts  from  given  values  x,  y  and  cal- 
culates the  change  of  y  corresponding  to  a  certain  small  change 
of  x.  Let  h  be  the  change  of  x  and  k  the  change  of  y,  so  that 
x  +  h,  y  +  k  are  the  coordinates  of  a  point  on  the  curve  satisfying 
the  differential  equation  and  passing  through  the  point  x,  y.  k  is 
calculated  in  the  following  manner.  We  calculate  in  succession 
four  values  ki,  k2,  k3,  &4  by  the  following  equations  — 


DIFFERENTIAL  AND   INTEGRAL   CALCULUS. 


133 


fa  =  f(z,  y)h, 

fa  = 


fa  =  flx  +  2>y  +  ^jh, 
fa=f(x+h,y+fa)h. 
We  then  form  the  arithmetical  means 


fa  fa  +  fa 

—     and        *"~~~» 


and  find  with  a  high  degree  of   approximation  as  long  as  h  is 
not  too  large 

k  =  p+$(q-  p).1 

The  new  values 

X  =  x+h,     Y=y  +  k 

are  then  substituted  for  x  and  y  and  in  the  same  way  the  coordi- 
nates of  a  third  point  are  calculated  and  so  on. 

This  calculation  may  be  performed  graphically  in  a  profitable 
manner,  if  the  function  f(x,  y)  is  represented  in  a  way  suited  to 


FIG.  91. 
1  See  W.  Kutta,  Zeitschrift  fur  Malhematik  und  Physik,  Vol.  46,  p.  443. 


134  GRAPHICAL   METHODS. 

the  purpose.  Let  us  suppose  a  number  of  equidistant  parallels 
to  the  axis  of  ordinates  :  x  =  XQ,  x  =  x\,  x  =  x2,  x  =  x%, 
Along  these  lines  f(x,  y)  is  a  function  of  y.  Let  us  lay  off  the 
values  of  f(x,  y)  as  ordinates  to  the  abscissa  y,  the  axis  of  y  being 
taken  as  the  axis  of  abscissas.  We  thus  obtain  a  number  of 
curves  representing  the  functions  f(x0,  y),  /(arlf  y),  f(xz,  y), 
Starting  from  a  point  A(x0,  yQ}  on  the  first  vertical  x  =  XQ  (Fig. 
91)  we  proceed  to  a  point  B:  on  the  vertical  x  =  x2  in  the  following 
way.  By  drawing  a  horizontal  line  through  A  we  find  the 
point  A'  on  the  curve  representing  f(x0,  y).  Its  ordinate  is  equal 
to  f(x0,  i/o).  Projecting  the  point  A'  onto  the  axis  of  x  we  find  A" 
and  draw  the  line  PA".  P  is  a  point  on  the  negative  side  of 
the  2/-axis  and  PO  is  equal  to  the  unit  of  length  by  which  the 
lines  representing  f(x,  y)  are  measured.  Thus 

OA"IPO  =  f(xQ,  2/0). 

Now  we  draw  ABi  perpendicular  to  PA",  so  that  if  h  and  /,'i 
denote  the  differences  of  the  coordinates  of  A  and  B,  we  have 

kjh  =  OA"/PO, 
h  =  f(xQ,  yo)h. 

From  Ci  the  point  of  intersection  of  the  line  AE\  and  the  vertical 
x  =  x\  we  find  C\  and  C\"  in  the  same  way  as  we  found  A'  and  A" 
from  A,  only  that  C\  is  taken  in  the  curve  representing  the 
values  of  f(xi,  y},  and  draw  the  line  AB2  perpendicular  to  PC\". 
Denoting  the  difference  of  the  ordinates  of  A  and  B2  by  A-2  we  have 

h     Od" 

h  =  -po= 

or 


From  Cz  the  point  of  intersection  of  the  line  AB%  and  the 
vertical  x  =  x\  we  find  in  the  same  way  a  point  B$  on  the  vertical 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS.  135 

x  =  x2  and  the  difference  &3  between  the  ordinate  of  B3  and  that 
of  A  is 


From  j?3  we  pass  horizontally  to  B3'  on  the  curve  representing 
f(x2y)  and  vertically  down  to  J53".  The  line  AB±  is  then  drawn 
perpendicular  to  PB3",  so  that  the  difference  &4  between  the 
ordinates  of  B^  and  A  is 


The  bisection  of  #2^3  and  of  -Z?il?4  gives  us  the  points  E\  and  £2 
and  the  point  B  is  taken  between  EI  and  £2,  so  that  its  distance 
from  £"1  is  half  its  distance  from  £2.  The  point  B  is  with  a  high 
degree  of  approximation  a  point  of  the  curve  that  passes  through 
A  and  satisfies  the  differential  equation. 

B  is  then  taken  as  a  new  point  of  departure  instead  of  A,  and 
in  this  manner  a  series  of  points  of  the  curve  are  found. 

In  order  to  get  an  idea  of  the  accuracy  attained  the  distance 
of  the  vertical  lines  is  altered.  For  instance,  we  may  leave  out 
the  verticals  x  =  x\  and  x  =  x3,  and  reach  the  point  on  the 
vertical  x  =  x^  in  one  step  instead  of  two.  The  error  of  this 
point  should  then  be  about  sixteen  times  as  large  as  the  error  on 
the  same  vertical  reached  by  two  steps,  so  that  the  error  of  the 
latter  should  be  about  one-fifteenth  of  the  distance  of  the  two. 
If  their  distance  is  not  appreciable  the  smaller  steps  are  evidently 
unnecessarily  small. 

The  values  of  f(x,  y)  may  become  so  large  that  an  incon- 
veniently small  unit  of  length  must  be  applied  to  plot  them.  In 
this  case  x  and  y  have  to  change  parts  and  the  differential 
equation  is  written  in  the  form 

dx          1 


dy      f(x,  y) ' 
The  values  of  l/f(x,  y)  are  then  plotted  for  equidistant  values  of 


136  GRAPHICAL   METHODS. 

y  as  ordinates  to  the  abscissa  x  and  the  constructions  are  changed 
accordingly. 

§  16.  Differential  Equations  of  the  Second  and  Higher  Orders.  — 
Differential  equations  of  the  second  order  may  be  written  in  the 
form 


'dx' 

Let  us  introduce  the  radius  of  curvature  instead  of  the  second 
differential  coefficient.  Suppose  we  pass  along  a  curve  that 
satisfies  the  equation  and  the  direction  of  our  motion  is  deter- 
mined by  the  angle  a  it  forms  with  the  positive  axis  of  x  (counted 
in  the  usual  way  from  the  positive  axis  of  x  through  ninety 
degrees  to  the  positive  axis  of  y  and  so  on),  s  being  the  length  of 
the  curve  counted  from  a  certain  point  from  which  we  start. 
We  then  have 

dy  dx 

•y-  =  tg  a,      -j-  —  COS  a. 
dx  ds 

Consequently 

d2y  _      1         da          1         da 

dx2      cos2  a    dx       cos3  a    ds  ' 
or 

da  d*y 

T  -**«»• 

da/ds  measures  the  "curvature,"  the  rate  of  change  of  direction 
as  we  pass  along  the  curve,  counted  positive  when  the  change 
takes  place  to  the  side  of  greater  values  of  a  (if  the  positive  axis 
of  x  is  drawn  to  the  right  and  the  positive  axis  of  y  upwards  a 
positive  value  of  da/ds  means  that  the  path  turns  to  the  left). 
Let  us  count  the  radius  of  curvature  with  the  same  sign  as  da/ds 
and  let  us  denote  it  by  p.  Then  we  have 

-  =  cos3a/(z,  y,tg«). 

Thus  the  differential  equation  of  the  second  order  may  be  said 
to  give  the  radius  of  curvature  as  a  function  of  x,  y,  a,  that  is  to 
say,  as  a  function  of  place  and  direction. 


DIFFERENTIAL    AND    IXTEGEAL    CALCULUS.  137 

Let  us  assume  that  this  function  of  three  variables  is  repre- 
sented by  a  diagram,  so  that  the  length  and  sign  of  p  may  quickly 
be  obtained  for  any  point  and  any  direction. 

Starting  from  any  given  point  in  any  given  direction  we  can 
then  approximate  the  curve  satisfying  the  differential  equation 
by  a  series  of  circular  arcs.  Let  A  (Fig.  92)  be  the  starting  point. 
We  make  MaA  perpendicular  to  the  given  direction  and  equal  to 
p  in  length.  For  positive  values  of  p,  Ma  must  be  on  the  positive 
side  of  the  given  direction,  for  negative 
values  on  the  negative  side.  Ma  is 
the  center  of  curvature  for  the  curve 
at  A.  With  Ma  as  center  and  MaA 
as  radius  we  draw  a  circular  arc  AB 
and  draw  the  line  BMa.  On  this  line 
or  on  its  production  we  mark  the  . 
point  Mb  at  a  distance  from  B  equal 
to  the  value  of  p  that  corresponds  to 
B  and  to  the  direction  in  which  the  p^,.  92. 

circular  arc  reaches  B.     With  3/b  as 
center  and  MiB  as  radius  we  draw  a  circular  arc  BC  and  so  on. 

The  true  curve  changes  its  radius  of  curvature  continuously, 
while  our  approximation  changes  it  abruptly  at  the  points 
A,  B,  C,  -  •  -  .  The  smaller  the  circular  arcs  the  less  will  accu- 
rately-drawn circular  arcs  deviate  from  the  curve.  But  it  must 
be  kept  in  mind  that  small  errors  cannot  be  avoided,  when 
passing  from  one  arc  to  the  next.  Hence,  if  the  arcs  are  taken 
very  small  so  that  their  number  for  a  given  length  of  curve 
increases  unduly,  the  accuracy  will  not  be  greater  than  with 
somewhat  longer  arcs.  The  best  length  cannot  well  be  defined 
mathematically;  it  must  be  left  to  the  experience  of  the  draughts- 
man. 

Some  advantage  may  be  gained  by  letting  the  centers  and  the 
radii  of  the  circular  arcs  deviate  from  the  stated  values.  The 
circular  arc  AB  (Fig.  92)  is  evidently  drawn  with  too  small  a 
radius  because  the  radius  of  the  curve  increases  towards  B.  If 


138  GRAPHICAL   METHODS. 

we  had  taken  the  radius  equal  to  MbB  it  would  have  been  too 
large.  A  better  approximation  is  evidently  obtained  by  making 
the  radius  of  the  first  circular  arc  equal  to  the  mean  of  MaA  and 
MbB,  and  the  direction  with  which  it  reaches  B  will  also  be  closer 
to  the  right  direction. 

To  facilitate  the  plotting  an  instrument  may  be  used  consisting 
of  a  flat  ruler  with  a  hole  on  one  end  for  a  pencil  or  a  capillary 
tube  or  any  other  device  for  tracing  a  line.  A  straight  line 
with  a  scale  is  marked  along  the  middle  of  the  ruler  and  a  little 
tripod  of  sewing  needles  is  placed  with  one  foot  on  the  line  and 
two  feet  on  the  paper.  Thus  the  pencil  traces  a  circular  arc. 
When  the  radius  is  changed,  the  ruler  is  held  in  its  position  by 
pressing  it  against  the  paper  until  the  tripod  is  moved  to  a  new 
position.  By  this  device  the  pencil  must  continue  its  path  in 
exactly  the  same  direction,  while  with  the  use  of  ordinary  com- 
passes it  is  not  easy  to  avoid  a  slight  break  in  the  curve  at  the 
joint  of  two  circular  arcs. 

Another  method  consists  in  a  generalization  of  the  method 
for  the  graphical  solution  of  a  differential  equation  of  the  first 
order. 

A  differential  equation  of  the  second  order 


may  be  written  in  the  form  of  two  simultaneous  equations  of  the 
first  order: 

dy 

_  ^L       —       ty 

dx~Z' 


Let  us  consider  the  more  general  form,  in  which  the  differential 
coefficients  of  two  functions  y,  z  of  x  are  given  as  functions  of 
x,  y,  z: 

I  -MM. 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS.  139 

~  =  g(x,y,z). 

We  may  interpret  x,  y,  z  as  the  coordinates  of  a  point  in  space 
and  the  differential  equation  as  a  law  establishing  a  certain 
direction  or  the  opposite  at  every  point  in  space  where  f(x,  y,  2) 
and  g(x,  y,  2)  are  defined.  A  curve  in  space  satisfies  the  dif- 
ferential equation,  when  it  never  deviates  from  the  prescribed 
direction.  Its  projection  in  the  xy  plane  represents  the  function 
y  and  its  projection  in  the  xz  plane  represents  the  function  z. 

Let  us  represent  y  and  z  as  ordinates  and  x  as  abscissa  in  the 
same  plane  with  the  same  system  of  coordinates.  Any  point  in 


FIG. 


space  is  represented  by  two  points  with  the  same  abscissa.  The 
functions  f(x,  y,  z)  and  g(x,  y,  z)  we  suppose  to  be  given  either 
by  diagrams  or  by  certain  methods  of  construction  or  calculation. 
For  any  point  that  we  have  to  deal  with,  the  values  of  f(x,  y,  2) 
and  g(x,  y,  2)  are  plotted  as  ordinates  to  the  abscissa  x,  but  for 
clearness  sake  not  in  the  same  system  of  coordinates  as  y  and  2, 
but  in  another  system  with  the  same  axis  of  ordinates  and  an 
axis  of  x  parallel  to  the  first  and  removed  far  enough  so  that  the 
drawings  in  the  two  systems  do  not  interfere  with  one  another. 


140  GRAPHICAL   METHODS. 

Starting  from  a  certain  point  P(xp,  yp,  zp)  in  space  we  represent 
it  by  the  two  points  PI(XP,  yp}  and  PZ(XP,  zp)  in  the  first  system 
and  the  values  of  f(xp,  yp,  zp)  and  g(xp,  yp,  zp)  by  the  two  points 
AI  and  AZ  in  the  second  system  of  coordinates  (Fig.  93).  The 
points  AI  and  AZ  determine  certain  directions  MA\,  and  MAz' 
of  the  curves  x,  y  and  x,  z,  the  point  M  (Fig.  93)  being  placed  at  a 
distance  from  the  axis  of  ordinates  equal  to  the  unit  of  length  by 
which  the  ordinates  representing  f(x,  y,  z)  and  g(x,  y,  z)  are 
measured.  Through  PI  and  P2  we  draw  parallels  to  MAi  and 
MAz  as  far  as  Qi  and  Qz  with  the  coordinates  xq,  yq  and  xq,  zq. 
With  these  coordinates  the  values  f(xq,  yq,  zq)  and  g(xq,  yq,  zq) 
are  determined,  which  we  represent  by  the  ordinates  of  the 
points  BI,  Bz.  These  points  again  determine  certain  directions 
parallel  to  which  the  lines  QiRi  and  QzRz  are  drawn,  etc.  In  this 
manner  we  find  first  approximations  y\  and  Zi  for  the  functions 
y  and  z  and  corresponding  to  these  approximations  we  find 
curves  representing  f(x,  yi,  Zi)  and  g(x,  y\,  Zi).  These  curves  are 
now  integrated  graphically,  the  integral  curve  of  f(x,  yi,  zO 
beginning  at  PI  and  the  integral  curve  of  g(x,  y\,  Zi)  at  P2  and 
lead  to  second  approximations  yz  and  ZQ  '• 


=  yP  +  I  f(x,  yi,  zjdx 
J*P 

=  zp  +  I  g(x,  yi*  *i)d 
" 


For  these  second  approximations  the  values  of  f(x,  y^,  Za)  and 
g(x,  y^,  %z)  are  determined  at  a  number  of  points  along  the  curves 
x,  7/2  and  x,  Zz  sufficiently  close  to  construct  the  curves  representing 
f(x,  yz,  Zi)  and  g(x,  y2,  Zz).  By  their  integration  a  third  approxi- 
mation 2/3,  Zs  is  obtained 

2/3  =  yP  +   I    f(x,  yz,  *2)dx, 
JXP 

=  zp+   I    g(x,  yz,  zz)dx, 


r 


DIFFERENTIAL   AND    INTEGRAL   CALCULUS.  141 

and  so  on  as  long  as  a  deviation  of  an  approximation  from  the 
one  before  can  still  be  detected.  As  soon  as  there  is  no  deviation 
for  a  certain  distance  x  —  xp  the  curve  represents  the  true  solu- 
tion (as  far  as  the  accuracy  of  the  drawing  goes).  The  curve  is 
continued  by  taking  its  last  point  as  a  new  starting  point  for  a 
similar  operation. 

The  distance  over  which  the  integral  is  taken  can  in  general 
not  surpass  a  certain  limit  where  the  convergence  of  the  approxi- 
mations ceases.  But  we  are  free  to  make  it  as  small  as  we  please 
and  accordingly  increase  the  number  of  operations  to  reach  a 
given  distance.  It  is  evidently  not  economical  to  make  it  too 
small.  On  the  contrary,  we  shall  choose  it  as  large  as  possible 
without  unduly  increasing  the  number  of  approximations. 

In  the  case  of  a  differential  equation 

j?-f  (**:£, 

we  have  f(x,  y,  z)  =  z,  and  the  curve  z,  x  is  identical  with  the 
curve  representing  the  values  of  f(x,  y,  z).  We  shall  therefore 
draw  it  only  once. 

The  proof  of  the  convergence  of  the  approximations  is  almost 
the  same  as  in  the  case  of  the  differential  equation  of  the  first 
order. 

For  the  n  +  1st  approximation  we  have 

yn+l  =  yp  +  J   f(x,  yn,  zn)dx;    zn+i  =  zp  +  J    g(x,  yn,  zn}dx. 

For  the  true  curve  that  passes  through  the  point  xp,  yp,  zp  we 
find  by  integration 


y 

hence 


£r 
f(x,  y,  z)dx\    z  =  zp  +    I    g(x,  y,  z)dx; 
Jrf 

yn+i  -  y  =    I    [f(x,  Vn,  z»)  -  f(x,  y,  z)]dx; 

" xp' 

zn+i  -  z  =  J    [g(x,  yn,  zn)  -  g(x,  y,  z)]dx. 


142  GRAPHICAL   METHODS. 

Now  let  us  write 

-,  ,       f(          s       f(x  - 

/(*,  yn,  zn}  -  f(x,  y,  z)  =  - 


yn  —  y 

f(x,  y,  zn)  -  f(x,  y,  z)        _ 

zn       z 
and  similarly 

g(x,  yn,  zn)  —  </(#,  y,  zn) 

yn  —  y 

g(x,y,zn}-g(x,y,z) 

Zn-z          ~  (Zn  ~  2)* 
The  quotients  of  differences 

/fcy»,g.)-/(*,y,g.) 

and  the  three  others  are  equal  to  certain  values  of  df/dy,  df/dz, 
dg/dy,  dg/dz  for  values  of  y,  z  between  y  and  yn  and  between  z 
and  zn  (y,  yn,  z,  zn  not  excluded).  Let  us  assume  that  for  the 
region  of  all  the  values  of  x,  y,  z  concerned  the  absolute  value  of 
df/dy  and  dffdz,  is  not  greater  than  MI,  and  that  of  dg/dy  and 
dg/dz  not  greater  than  3/2,  and  that  dn,  en  denote  the  maximum 
of  the  absolute  values  of  y  —  yn  and  z  —  zn  in  the  interval 
xp  to  x.  Then  it  follows  that  the  absolute  values  of 

f(x,  yn,  zn)  —  f(x,  y,  z)     and     g(x,  yn,  zn)  —  g(x,  y,  z) 
are  not  greater  than 

Hence  for  the  maximum  values  of  yn+i  —  y  and  zn+i  —  z,  which 
are  denoted  by  5n+1  and  e^+i  we  obtain  the  limits 

and 

8n+i  +  en+i  ^  (3/i  +  3/2)  |  x  —  xp    (8n  +  cn). 

If  therefore  the  interval  x  —  xp  of  the  integration  is  so  far 
reduced  that 

(3/i  +  3/2)  |  x  —  xp  |  ^  k  <  1, 


DIFFERENTIAL   AND   INTEGRAL    CALCULUS.  143 

5n+i  +  en+i  is  not  larger  than  the  fraction  k  of  (8n  +  en),  but 
from  the  same  reason 

(5B  +  en)  ^  ^(Sn-!  +  €„_!),      (5n_!  +  en_t)  <  &(5n_2  +  6^2),  etc.; 

therefore 

5n+1  +  en+1  ^  &"(5i  +  ei). 

That  is  to  say,  for  a  sufficiently  large  value  of  n  8n+i  and  en+i 
will  both  become  as  small  as  we  please. 

As  in  the  case  of  the  differential  equation  of  the  first  order  it  is 
not  worth  while,  as  a  rule,  to  investigate  the  convergence  for  the 
purpose  of  finding  a  sufficiently  close  approximation  by  graphical 
methods.  It  is  better  at  once  to  tackle  the  task  of  drawing  the 
approximations  and  to  repeat  the  operations  until  no  further 
improvement  is  obtained.  The  curve  will  then  satisfy  the 
differential  equation  as  far  as  the  graphical  methods  allow  it 
to  be  recognized. 

^Yhen  the  values  of  f(x,  y,  z)  or  g(x,  y,  z)  become  too  large 
we  can  have  recourse  to  the  same  device  that  we  found  useful 
with  the  differential  equation  of  the  first  order.  Instead  of  x, 
one  of  the  other  two  variables  y  or  z  may  be  considered  as  inde- 
pendent, so  that  the  equations  take  the  form 

dx  _         1  dz  _  g(x,  y,  z) 

dy  ~  f(x,  y,  z)'  dy~  f(x,  y,  z) ' 
or 

dx  _         I  dy  _  f(x,  y,  z) 

dz  ~  g(x,  y,  z) '  dz       g(x,  y,  z) ' 

or  we  may  introduce  a  new  system  of  coordinates  xf,  y',  z'  and 
consider  the  resulting  differential  equations. 

The  second  method  for  the  integration  of  differential  equations 
of  the  first  order  can  also  be  generalized  to  include  the  second 
order.  Let  us  again  consider  the  more  general  case 


Starting  from  a  point  x,  y,  z  the  changes  of  y  and  z  (denoted  by 


144  GRAPHICAL    METHODS. 

k  and  /)  can  be  calculated  for  a  small  change  A  of  z  by  the  fol- 
lowing formulas  analogous  to  those  used  for  one  differential 
equation  of  the  first  order: 

ki  =  /(#,  y,  z)A;  l\  =  g(x,  y,  z)h; 

fc  =  /(*+  \ ,  y+  | ,  ^+  |)  A;    fe  =  ^  (.r+  f~,  y+ 1 ,  2+  |)A; 


:>• 


*4  =  /(a:  +  A,  y+  £3,  2  +  ^s)A;      k  =  flf(ar  +  A,  y  +  h,  z  +  I3)h; 

h  +  h          k,  +  h  ,     k  +  k          h  +  k 

P  =       2       '  q  =       2      ;          P  =       2      '  q  =      2     ' 

and  with  a  high  degree  of  approximation, 


These  calculations  may  be  performed  graphically.  For  this 
purpose  the  functions  f(x,  y,  z)  and  g(x,  y,  z)  must  be  given  in 
some  handy  form.  We  notice  that  in  our  formulas  the  first 
argument  assumes  the  values  x,  x  +  A/2,  x  +  h.  In  the  next 
step  where  x  -\-  h,  y  -{-  k,  z  -\-  I  are  the  coordinates  of  the  starting 
point  that  play  the  same  part  that  x,  y,  z  played  in  the  first 
step,  we  are  free  to  make  the  change  of  the  first  argument  the 
same  as  in  the  first  step,  so  that  in  the  formulas  of  the  second 
step  it  assumes  the  values  x  +  h,  x  +  f  A,  x  +  2h  and  so  on  for 
the  following  steps.  All  the  values  of  the  first  argument  can 
thus  be  assumed  equidistant.  Let  us  denote  these  equidistant 
values  by 

.TO,  xi,  x2,  x3, 

The  values  of  f(x,  y,  z)  and  g(x,  y,  z)  appear  in  all  our  formulas 
only  for  the  constant  values 

X  =   X0,  Xi,  X2, 

For  each  of  these  constants  /  and  g  are  functions  of  two  inde- 
pendent variables  and  as  such  may  be  represented  graphically 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


145 


by  drawings  giving  the  curves  /  =  const,  and  g  =  const.,  each 
value  of  x  corresponding  to  a  separate  drawing.  These  drawings 
we  must  consider  as  the  graphical  form  in  which  the  differential 
equations  are  given.  It  may  of  course  sometimes  be  very  tire- 
some to  translate  the  analytical  form  of  a  differential  equation 
into  a  graphical  form,  but  this  trouble  ought  not  to  be  laid  to 
the  account  of  the  graphical  method. 

The  method  now  is  similar  to  that  used  for  the  differential 
equation  of  the  first  order,  y  and  z  are  plotted  as  ordinates  in 
the  same  system  in  which  x  is  the  abscissa.  Equidistant  parallels 
to  the  axis  of  ordinates  are  drawn 


x  = 


etc. 


x  =  x\,  x  — 

On  the  first  x  =  XQ  we  mark  two  points  with  ordinates  yo  and  ZQ, 
and  from  the  drawing  that  gives  the  values  of  f(xo,  y,  z)  and 


Fia.  94. 

g(x0,  y,  z)  as  functions  of'  y  and  z  we  read  the  values  f(x0,  y0,  z0) 
and  gfa,  y0,  z0)  and  draw  the  lines  from  x0,  y0,  and  x0,  z0  to  the 
points 

£2,  yo  +  h    and    z2,  z0  +  h. 

The  intersections  of  these  lines  with  the  parallel  x  =  x\  furnishes 
the  points 
11 


146  GKAPHICAL   METHODS. 


, 

xi,  2/o  +  -^     and 
With  these  ordinates  we  find  from  the  second  drawing  the  values 

.(         ,h         k\  (          ,h       ,k\ 

f  [  xi,  yo  +  2",  20  +  2  )     and    0  I  *i»  2/o  +  -j  ,  *o  +  2  )  > 

and  by  their  help  we  can  draw  the  lines  from  XQ,  yo  and  XQ,  ZQ 
to  the  points 

and 


The  intersections  of  these  lines  with  the  line  x  =  xi  furnishes  the 
points 

*i,  yo  +  ^     and    ari,  z0  +  2  > 

and  with  these  ordinates  we  find  the  values 

./  ,fc         .    4\         /  ,    fe 

/  I  *i,  2/o  +  ^-,  20  H-  g   )  ,     f7  1  a-i,  2/o  +  -^  , 

which  enable  us  to  draw  the  lines  from  XQ,  yo  and  XQ,  ZQ  to  Xz, 
yo  +  h  and  a%,  ZQ  +  13. 

With  these  two  ordinates  we  find  from  the  third  diagram  (a;  =  #2) 
the  values 

/(a&,  yo  +  ^3,  20  rf  4)     and     gfa,  y0  +  Ar3,  z0  +  4), 

which  finally  enable  us  to  draw  the  lines  from  x0yo  and  a-oZo  to 
%,  yo  +  h  and  x2,  z0  +  ^4. 

On  the  vertical  line  x  —  Xz  we  thus  obtain  four  points,  BI,  B2, 
B3,  #4,  corresponding  to  y0  +  ki,  y0  +  fa,  yo  +  k3,  y0  +  A:4  and 
four  points,  BI,  B2',  B3',  B*,  corresponding  to  z0  +  li,  ZQ  +  k, 
zo  +  k,  20  +  k  (Fig.  94). 

5253  and  5iB4  are  bisected  by  the  points  d  and  C2;  52'53' 
and  Bi'Bt  by  the  points  d',  C2'.  Finally  dC2  and  Ci'C2  are 
divided  into  three  equal  parts  and  the  points  B  and  E'  are  found 
in  the  dividing  points  nearest  to  Ci  and  C\. 

The  same  construction  is  then  repeated  with  B  and  B'  as 
starting  points  and  furnishes  two  new  points  on  the  vertical 


DIFFERENTIAL   AND   INTEGRAL    CALCULUS.  147 

x  =  0-4  and  so  on.  To  test  the  accuracy  the  construction  is 
repeated  with  intervals  of  x  of  double  the  size.  The  difference 
in  the  values  of  y  and  of  2  found  for  x  =  x^  enables  us  to  estimate 
the  errors  of  the  first  construction — they  are  about  one-fifteenth 
of  the  observed  differences. 

Both  methods  are  without  difficulty  generalized  for  the  integra- 
tion of  differential  equations  of  any  order.  We  can  write  a 
differential  equation  of  the  nth  order  in  the  form 


or  in  the  form  of  n  simultaneous  equations  of  the  first  order 

dx 

dt  =  Xl> 

dx, 


fa 


f(t,  X, 


A  more  general  and  more  symmetrical  form  is 

dx 

-    =  fi(t,  x,  xi,  -  •  •  zn_i), 


The  functions  x,  Xi,  a&,  •  •  •  a:«_i  are  then  represented  as  ordinates 
to  the  abscissa  i,  so  that  we  have  n  different  curves.  When  the 
function  f(t,  x,  xi,  Xz,  —  -  x^i)  is  given  in  a  handy  form,  so  that 


148  GRAPHICAL   METHODS. 

its  value  may  be  quickly  found  for  any  given  values  of  t,  x,  x\, 
•  •  -  Zn_i,  there  is  no  difficulty  in  constructing  n  curves  whose 
ordinates  represent  the  functions  x,  xi,  Xz,  —  -  xn-i.  Starting 
from  given  values  of  t,  x,  xi,  x%,  •  •  •  xn-i  we  have  only  to  apply 
the  same  methods  that  have  been  explained  for  the  first  and  the 
second  order. 


COLUMBIA  UNIVERSITY   PRESS 

Columbia  University  in  the  City  of  New  York 


The  Press  was  incorporated  June  8,  1893,  to  promote  the  publication 
of  the  results  of  original  research.  It  is  a  private  corporation,  related  di- 
rectly to  Columbia  University  by  the  provisions  that  its  Trustees  shall  be 
officers  of  the  University  and  that  the  President  of  Columbia  University 
shall  be  President  of  the  Press. 


The  publications  of  the  Columbia  University  Press  include  works  on 
Biography,  History,  Economics,  Education,  Philosophy,  Linguistics,  and 
Literature,  and  the  following  series  : 

Columbia  University  Contributions  to  Anthropology. 

Columbia  University  Biological  Series. 

Columbia  University  Studies  in  Cancer  and  Allied  Subjects. 

Columbia  University  Studies  in  Classical  Philology. 

Columbia  University  Studies  in  Comparative  Literature. 

Columbia  University  Studies  in  English. 

Columbia  University  Geological  Series. 

Columbia  University  Germanic  Studies. 

Columbia  University  Indo-Iranian  Series. 

Columbia  University  Contributions  to  Oriental  History  and 
Philology. 

Columbia  University  Oriental  Studies. 

Columbia  University  Studies  in  Romance  Philology  and  Liter, 
atnre. 

Adams  Lectures.  Carpentiw  Lectures. 

Julius  Beer  Lectures.  Hewitt  Lectures. 

Blumenthal  Lectures.  Jesup  Lectures. 

Catalogues  will  be  sent  free  on  application. 


LEMCKE  &  BUECHNER,  Agents 

30-32  WEST  a7th  ST.,   NEW  YORK 


COLUMBIA  UNIVERSITY  PRESS 

Columbia  University  in  the  City  of  New  York 
COLUMBIA  UNIVERSITY   LECTURES 


ADAMS  LECTURES 

Graphical  Methods.  By  CARL  RUNGE,  Ph.D.,  Professor  of 
Applied  Mathematics  in  the  University  of  Gottingen  ;  Kaiser 
Wilhelm  Professor  of  German  History  and  Institutions  for  the 
year  1909-1910.  8vo,  cloth,  pp.  ix-j-148.  Price,  $1.50  net. 

JULIUS  BEER  LECTURES 

Social  Evolution  and  Political  Theory.  By  LEONARD  T. 
HOBHOUSE,  Professor  of  Sociology  in  the  University  of  London. 
12mo,  cloth,  pp.  ix-f  218.  Price,  $1.50  ret. 

BLUMENTHAL  LECTURES 

Political  Problems  of  American   Development.      By 

ALBERT  SHAW,  LL.D.,  Editor  of  the  Review  of  Reviews.     12mo, 

cloth,  pp.  vii+268.    Price,  $1.50  net. 
Constitutional  Government  in  the  United  States.    By 

WOODROW  WILSON,  LL.D.,  President  of  Princeton  University. 

12mo,  cloth,  pp.  vii+236.     Price,  $1.50  net. 
The  Principles  of  Politics  from  the  Viewpoint  of  the 

American  Citizen.     By  JEREMIAH  W.  JENKS,  LL.D.,  Pro- 
fessor of  Political  Economy  and  Politics  in  Cornell  University. 

12mo,  cloth,  pp.  xviii+187.    Price,  $1.50  net. 
The  Cost  of  Our  National   Government.     By  HEKRY 

JONES  FORD,   Professor  of    Politics    in    Princeton    University. 

12mo,  cloth,  pp.  xv+147.     Price,  $1.50  net. 
The  Business  of  Congress.    By  HON.  SAMUEL  W.  MCCALL, 

Member  of  Congress  for  Massachusetts.     12mo,  cloth,  pp.  vii+ 

215.     Price,  $1.50  net. 

CARPENTIER  LECTURES 

The  Nature  and  Sources  of  the  Law.  By  JOHN  CHIPMAN 
GRAY,  LL.D.,  Eoyall  Professor  of  Law  in  Harvard  University. 
12mo,  cloth,  pp.  xii+332.  Price,  $1.50  net. 

World  Organization  as  Affected  by  the  Nature  of  the 
Modern  State.  By  HON.  DAVID  JAYNE  HILL,  American 
Ambassador  to  Germany.  12mo,  cloth,  pp.  ix+214.  Price, 
$1.50  net. 

The  Genius  of  the  Common  Law.  By  the  KT.  HON.  SIR 
FREDERICK  POLLOCK,  Bart.,  D.C.L.,  LL.D.,  Bencher  of  Lincoln's 
Inn,  Barrister-at-Law.  12mo,  cloth,  pp.  vii+141.  Price,  $1.50 
net. 

LEMCKE  &   BUECHNER,  Agents 

30-32  West  27th  Street,  New  York 


COLUMBIA  UNIVERSITY  PRESS 

Columbia  University  in  the  City  of  New  York 
COLUMBIA  UNIVERSITY   LECTURES 


HEWITT  LECTURES 

The  Problem  of  Monopoly.  By  JOHN  BATES  CLARK,  LL.D., 
Professor  of  Political  Economy,  Columbia  University.  12mo, 
cloth,  pp.  vi+128.  Price,  $  1.50  net. 

Power.  By  CHARLES  EDWARD  LUCRE,  Ph.D.,  Professor  of  Mechan- 
ical Engineering,  Columbia  University.  12mo,  cloth,  pp.  vii+ 
316.  Illustrated.  Price,  $2.00  net. 

The  Doctrine  of  Evolution.  Its  Basis  and  its  Scope.  By 
HENRY  EDWARD  CRAMPTON,  Ph.D.,  Professor  of  Zoology, 
Columbia  University.  12mo,  cloth,  pp.  ix+311.  Price,  $1.50  net. 

Medieval  Story  and  the  Beginnings  of  the  Social  Ideals 
of  English-Speaking-  People.  By  WILLIAM  WITHERLE 
LAWRENCE,  Ph.D  ,  Associate  Professor  of  English,  Columbia 
University.  12mo,  cloth,  pp.  xiv+236.  Price,  $1.50  net. 

JESUP  LECTURES 

Light.  By  KICHARD  C.  MACLAURIN,  LL.D.,  Sc.D.,  President  of  the 
Massachusetts  Institute  of  Technology.  12mo,  cloth,  pp.  ix-f  251. 
Portrait  and  figures.  Price,  $1.50  net. 

Scientific  Features  of  Modern  Medicine.  By  FREDERIC 
S.  LEE,  Ph.D.,  Dal  ton  Professor  of  Physiology,  Columbia  Uni- 
versity. 12mo,  cloth,  pp.  vi-f  183.  Price,  $1.50  net. 


Lectures  on  Science,  Philosophy  and  Art.  A  series  of 
twenty-one  lectures  descriptive  in  non-technical  language  of  the 
achievements  in  Science,  Philosophy  and  Art.  8vo,  cloth. 
Price,  $5.00  net. 

Lectures  on  Literature.  A  series  of  eighteen  lectures  on  liter- 
ary art  and  on  the  great  literatures  of  the  world,  ancient  and 
modern.  8vo,  cloth,  pp.  viii+404.  Price,  $2.00  net. 

Greek  Literature.  A  series  of  ten  lectures  delivered  at  Columbia 
University  by  scholars  from  various  universities,  in  the  spring  of 
1911.  8vo,  cloth,  pp.  vii+306.  Price,  $2. 00  net. 


LEMCKE  &  BUECHNER,  Agents 

30-32  West  27th  Street,  New  York 


1G    LIBRARY 


>,    ^>ffiGO'V      £E 


